The Great API Easter Egg Hunt!
APL+WIN
Works by starting in the centre and converges on the location by setting the upper or lower limit to the last value and adding or subtracting half the difference to the appropriate limit to the last guess on each iteration.
API                                                                               
id←¯20↑¯2↓GetUrl 'http://easter_egg_hunt.andrewfaraday.com/new_game'              
xh←yh←100 ⋄ xl←yl←0 ⋄ x←50 ⋄ y←50 ⋄ c←0                                           
:repeat                                                                           
    xy←GetUrl 'http://easter_egg_hunt.andrewfaraday.com/guess/',id,'/',(⍕x),'/',⍕y
    xy←(('higher'⍷xy)+(¯1×'lower'⍷xy)+2×'right'⍷xy)~0                             
    :if xy[1]=1 ⋄ xl←x ⋄ x←x+⌈(xh-x)÷2 ⋄ :endif                                   
    :if xy[1]=¯1 ⋄ xh←x ⋄ x←x-⌈(x-xl)÷2 ⋄ :endif                                  
    :if xy[2]=1 ⋄ yl←y ⋄ y←y+⌈(yh-y)÷2 ⋄ :endif                                   
    :if xy[2]=¯1 ⋄ yh←y ⋄ y←y-⌈(y-yl)÷2 ⋄ :endif                                  
    c←c+1                                                                         
:until 4=+/2↑xy                                                                   
'id:',id,' x:',(⍕x),' y:',(⍕y),' count:',⍕c 

This function uses the following function to do the API calls:
r←GetUrl url                                     
⎕wself←'HTTP' ⎕wi 'Create' 'MSXML2.ServerXMLHTTP'
⎕wi 'XOpen' 'GET' url 0                          
⎕wi 'XSend'                                      
r←⎕wi 'xresponseText'

The results for 10 trys were:
id:rbgprkxrqzzhwdfsbszn x:36 y:52 count:7      
id:nmpcxdqsdzhgrbtlcpbp x:35 y:49 count:6      
id:qqnsbpwnlbptxxblywnz x:99 y:22 count:6      
id:nsytnvcgnsyrgzvjcysc x:45 y:28 count:6      
id:yfkpfhphjpqxtqnwpmhv x:95 y:40 count:7      
id:kxhszzrhxqlnvwvwjgnm x:49 y:6 count:6       
id:rwnwfgdpzcjpzzfmgcfn x:93 y:34 count:7      
id:tcvhtpqlfrwngybsyzqh x:95 y:94 count:6      
id:pmlmqnprwcjggjfhttmy x:20 y:41 count:6      
id:kpsmmhfhxxrrlvbbgzkv x:9 y:28 count:5