I agree, finding a free cat usually means rescuing a stray, and there is no way to know if the cat has had his/her shots.
Your best bet is indeed a shelter cat. All the shelters I have ever encountered will only release a cat after it has had its shots, which they provide. You can always find a shelter cat that has already been fixed, as well. Shelter adoption fees can range from absurdly cheap (I tell my cat Midnight that he is the best $11 I ever spent!) to $50 and more.
A good place to start looking is on www.petfinder.com
which is a listing of animals in shelters all throughout the USA. Some are in SPCA's and Humane Societies, some are in no-kill shelters, some are in foster homes.
Simba will most likely be thrilled to get a companion. I know my cats love to sleep all in a bunch, they groom each other and play together. Occasionally, there is a little bit of what I call "kitty karate", but altogether they do get along very well.