So the standard recourse for such a situation is to request the vet to contact the adopting party and request that the cat be returned. If they will not return the cat, then it gets very complicated. If one is a foster provider through for example, the humane society, it is the humane society that will have to pursue that. It is called "legal standing" to bring an action. If one is fostering the pet for a relative, then that individual-the owner- would need to file suit.
There are 2 types of suits that can be brought
1. for economic damages - basically for the value of the pet, since many courts consider animals property, and will only allow for monetary damages
2. for specific performance- basically an order requiring the person who has possession of the animal to return the animal to its original owner. This is more complicated and will require 2 defendants, the vet and the person in possession (their identity would be revealed during discovery). It will come down to the particular judge's take on what is "just" - ie how long each party has the pet; if any party did any wrongdoing, etc. It is an equitable (versus statutory) remedy so each judge may have a different result.
Most of the legislatures have not revised codes to take into account society's changing attitudes towards our pets, and most still treat them as "chattel" or property, versus pets.
You may want to contact the humane society/animal control to see if they have an attorney they would recommend, basically one that would go the extra mile for a case such as this.