Animal control wouldn't even let you borrow a humane trap!? Maybe you can get one from another source... a vet maybe?
If you can trap them, you can take care of them yourself, if you have a quiet room to put them in. A big dog kennel for the mom and kits, some quiet music, and a calm, non-pushy human caretaker... Vet care (you might have to sedate the mom), flea treatments, worming, FIV/FLV testing (positive cats should be isolated from other cats, but can live a reasonably healthy life)... At least the kittens would have a safe place to grow up in.
Provide a hidey-hole for the mom. She'll want somewhere where she feels safe, if she's feral. And if she's stray, she'll still want to approach cautiously from a safe position.
Feral cats often look thin but healthy... they have adapted to their environment. So what you've got is either a feral that something has happened to--disease, accident, weather, maybe having to provide for kittens--to throw off its adaptations enough that it's now doing poorly. Strays are often thin and mangy-looking because they're not used to surviving on their own. So if your cat is healthy and her environment hasn't been too bad, she was likely owned by a human at some time--at least in kittenhood--or is a first-generation feral for whom socialization may be marginally possible (to the "I tolerate humans but don't let them touch me much" stage).
Do a web search on socializing feral kittens--it is possible.