Thomas took the bird to veterinarian because (its/it's) wing was broken. a. its b. it’s

The answer would be A.

The question you have to ask is, is it possessive? 
Yes it is possessive.
The wing is owned by the bird. Its wing was broken.

It's means it is or it has. 
It's would not make sense in this case because you can not put it is or it has into this sentence.

Iwould say ahope it