How can 蛙 be read as あ ("a") in Asui Tsuyu's name?
蛙 has ア【a】 as one of the on-yomi, and kanji compound is usually read in on-yomi (some also have a reading in kun-yomi, but in general, all of the kanji in the kanji compound are read in either kun-yomi only or on-yomi only, but not mixed).

Some example where 蛙 is read as "a":


  
  蛙黽 (abou): tree frog (toad)
  蛙鳴 (amei): frog calling​
  蛙鳴蝉噪 (ameisensou): annoying noise; fruitless argument; useless controversy
  


As for Asui, her kanji representation 蛙吹 is read in on-yomi as 蛙（ア） and 吹（スイ）