Is "biang" a phonologically possible syllable in Mandarin Chinese?
So time for an update…
If you want to play by the books, biang is not a permissible syllable. If you are concerned with what comes out of a speaker’s mouth, syllables like nim (contraction of 你們) are even possible, although they are technically surface realizations of a phonology that does not allow such syllables.
The surprising fact is that iang as a rime for the labial initials (b, p, ph, m) did not even exist in Middle Chinese, so the omission of biang is nothing new. That isn’t to say it couldn’t develop in Mandarin, but there was nothing shaping the language in that direction. The only likely source would be a borrowing sort of like how fiao is a borrowing from Wu, which has its own interesting story.
Old Answer
I’ve made a chart here for my own understanding of Standard Mandarin Chinese phonology a while ago:
        a       ai      au      an      aŋ      e       ə       əi      əu      ən      əŋ      i       ia      iai     iau     ian     iaŋ     ie      iə      iəu     iən     iəŋ     io      iu      iuan    iuə     iun     iuəŋ    aɻ      o       u       ua      uai     uan     uaŋ     uə      uəi     uən     uəŋ     m       n       ŋ      
        a       ai      ao      an      ang     ê       e       ei      ou      en      eng     yi      ya      yai     yao     yan     yang    yê      ye      you     yin     ying    yo      yu      yuan    yue     yun     yong    er      o       wu      wa      wai     wan     wang    wo      wei     wun     weng    m       n       ng     
p       ba      bai     bao     ban     bang                    bei             ben     beng    bi                      biao    bian    biang           bie     biu     bin     bing                                                            bo      bu                                                                                             
pʰ      pa      pai     pao     pan     pang                    pei     pou     pen     peng    pi                      piao    pian                    pie     piu     pin     ping                                                            po      pu                                                                                             
m       ma      mai     mao     man     mang            me      mei     mou     men     meng    mi                      miao    mian                    mie     miu     min     ming                                                            mo      mu                                                                                             
f       fa                      fan     fang                    fei     fou     fen     feng                            fiao                                                                                                                    fo      fu                                                                                             
t       da      dai     dao     dan     dang            de      dei     dou     den     deng    di                      diao    dian                    die     diu             ding                                                                    du                      duan            duo     dui     dun     dong                           
tʰ      ta      tai     tao     tan     tang            te              tou             teng    ti                      tiao    tian                    tie     tiu             ting                                                                    tu                      tuan            tuo     tui     tun     tong                           
n       na      nai     nao     nan     nang            ne      nei     nou     nen     neng    ni                      niao    nian    niang           nie     niu     nin     ning            nü              nüe                                     nu                      nuan            nuo                     nong                           
l       la      lai     lao     lan     lang            le      lei     lou             leng    li      lia             liao    lian    liang           lie     liu     lin     ling            lü              lüe                             lo      lu                      luan            luo             lun     long                           
ts      za      zai     zao     zan     zang            ze      zei     zou     zen     zeng    zi                                                                                                                                                      zu                      zuan            zuo     zui     zun     zong                           
tsʰ     ca      cai     cao     can     cang            ce              cou     cen     ceng    ci                                                                                                                                                      cu                      cuan            cuo     cui     cun     cong                           
s       sa      sai     sao     san     sang            se              sou     sen     seng    si                                                                                                                                                      su                      suan            suo     sui     sun     song                           
tʂ      zha     zhai    zhao    zhan    zhang           zhe     zhei    zhou    zhen    zheng   zhi                                                                                                                                                     zhu     zhua    zhuai   zhuan   zhuang  zhuo    zhui    zhun    zhong                          
tʂʰ     cha     chai    chao    chan    chang           che             chou    chen    cheng   chi                                                                                                                                                     chu     chua    chuai   chuan   chuang  chuo    chui    chun    chong                          
ʂ       sha     shai    shao    shan    shang           she     shei    shou    shen    sheng   shi                                                                                                                                                     shu     shua    shuai   shuan   shuang  shuo    shui    shun                                   
ɻ                       rao     ran     rang            re              rou     ren     reng    ri                                                                                                                                                      ru      rua             ruan            ruo     rui     run     rong                           
tɕ                                                                                              ji      jia             jiao    jian    jiang           jie     jiu     jin     jing            ju      juan    jue     jun     jiong                                                                                                                  
tɕʰ                                                                                             qi      qia             qiao    qian    qiang           qie     qiu     qin     qing            qu      quan    que     qun     qiong                                                                                                                  
ɕ                                                                                               xi      xia             xiao    xian    xiang           xie     xiu     xin     xing            xu      xuan    xue     xun     xiong                                                                                                                  
k       ga      gai     gao     gan     gang            ge      gei     gou     gen     geng                                                                                                                                                            gu      gua     guai    guan    guang   guo     gui     gun     gong                           
kʰ      ka      kai     kao     kan     kang            ke      kei     kou     ken     keng                                                                                                                                                            ku      kua     kuai    kuan    kuang   kuo     kui     kun     kong                           
x       ha      hai     hao     han     hang            he      hei     hou     hen     heng                                                                                                                                                            hu      hua     huai    huan    huang   huo     hui     hun     hong                           

The table contains every possible pinyin in Standard Mandarin Chinese sans tones. Columns are headed by the finals, and rows are headed by the initials. They are derived from the Standard Chinese phonology and Pinyin table pages on Wikipedia. A quick look at the table will reveal patterns in the phonology with a few oddities.
biang is certainly listed in the Pinyin table, but a quick search for any word with a pinyin of biang produces 0 results. And it makes sense: because the bilabial/labiodental series of initials (b-, p-, m-, f-) is excluded from combining with many finals (such as: -uan, -uo, -ui, -un, -ong). Have a look at the iaŋ column and you see that piang, miang, and fiang are missing; biang looks like it should not be there either.
Yes, there are oddities where a sound that looks like it should be there isn’t there (like bou, tei, and ten), but they can be easily explained as, “there weren’t any words that used that sound.” It’s what is called an “accidental gap,” and you can read more about it here: Phonotactics (âm pháp). It’s much more difficult to make a case for biang which exists in no dictionaries, is not in common usage, and is the sole proprietor of the Pinyin “biang.” Considering the facts, it’s my opinion that biang is illegal in Standard Mandarin Chinese phonology and was probably concocted for promotional purposes. What better way to stand out than to own a “rare” character, a “rare” sound, and an exclusive position in the Pinyin table?