One of my child sims found a sick stray cat and became good friends with it. He fed it and brushed it and everything, but the option to take to the vet never popped up. I thought I had read somewhere that you could adopt strays once the relationship bar was full enough, but I ended up eventually adding the cat to the household with cheats. THEN we went to the vet.
In real life people take strays to shelters. I think there should be an option to try and "catch" a stray and then take to the vet or give to the adoption agency.
