Try other organizations. When I was calling around, I found that different groups had different rules.
I like the one I'm with, and having other pets is no problem in being accepted as a foster home, especially because these homes are badly needed. They are fairly strict on what cats they will put in what situations. All their animals are tested and vaccinated (unless too young) before being put into a household. They will not put an FIV positive into a home with a negative, and visa versa.
There are been situations where they have stipulated that the animals be kept apart, and some where it wasn't an issue. Unfortunately, I have a 1-bedroom apt, so won't take animals that absolutely can't interract with my cats.
However, a mother and babies might cause some serious problems in a home with other cats unless they can be separated, because of the mother's protection instinct and the already-in-residence cats needing to protect their territory. That can be hairy. Two cats that normally would get along fine might end up going for the jugular when one has kittens.
I'm honest with the group I'm with as to what I feel i'm capable of and what I'm not, and I think they appreciate that. And I've since fostered trust with them because I do respect their rules.
If they won't let you foster because you have other cats, period, I'd go elsewhere. But if there are other reasons like the space involved, or just certain situations like medical or a mom with kittens, that you might find everywhere Ask more specific questions about what you're intersted in.
Interestingly, the group I foster for frowns upon their foster parents keeping any animals, and there is actually a large financial penalty. I'm not sure why they do that, maybe because of the financial investment they've made, or it keeps their fosters still having room to foster, one day I have to ask them about that.
Anyway, it is a wonderful, rewarding fun situation. Good luck!