pianistimmo,
Not everybody agrees with me on this issue but it is one I feel fairly strongly about.
We are morally obligated not to let an animal suffer needlessly. This comes with the territory - the rights and responsibilities of being a human.
Chipmunks generally get over a fright quickly. They are after all prey, and they are used to close calls. So I assume the cat actually injured it, maybe more than is obvious.
Now you have a chore on your hands that is unpleasant, but you must finish the job. You are known to be a soft hearted person and this is probably more repugnant to you than to many. I am not as obviously softhearted <ahem> but I feel the same way. Neverytheless it is the right thing to do. Put aside your squeamishness, think of the animal instead, and kill it as quickly and mercifully as possible.
If you can't (like my children. They would agree it is the right thing to do, but not be able to carry it out.) then find someone nearby to help.
Methods. Okay, this paragraph is gross. But giving you the task and not the tools is unfair. The cleanest and quickest way to do this is drop it in a paper bag and give the bag a hard smack against a wall or floor. It is instant and final. This is what I do for mice, and I NEVER let the kids see. Your other options are drowning and suffocation. Frankly I can't see you being able to stand doing that. It takes too long and gives you too much time to think about it.
Of course, you might be lucky and it will recover from being stunned. I've lived around a lot of chipmunks and never seen this happen, but we can hope. I would not give it back to the cat. Cats are anything but quick and merciful. I would also consider that as fast and alert as chipmunks are, they hardly ever get caught by a cat unless already sick or weakened.