As a general rule, you don't need to bring a crate you can just bring a collar and a leash. When I was a volunteer here my mother and I saw a couple of people taking their dogs on leashes to their cars. But if you adopt a bird, rabbit, cat or hamster you do have to bring your own crate so that the animal doesn't run away when his captors release him to you.
I believe you have to bring your own. I was not given a cage for my dog.
They have cardboard carriers for small animals at a cost of $5. They also have a thin leash that will get you home safe for $2. I recommend bringing your own. They also receive donated leashes from time to time and might have one for you, but it depends on availability.
Bring your own!
They will give you your new feline friend in a cardboard pet carrier that's like a folding jumbo lunchbox with carrying handles on top.
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