I'm currently making chicken stock, and I didn't have celery. Is it ok if I throw in the celery late (when my fiancee gets home with some)?

Of course.

Not to worry! The only necessary ingredients for chicken stock are chicken and water. Adding celery late, if you add it at all, could be an improvement over adding it early, as the flavor will be fresher.

I almost never have celery on hand and therefore I almost never put it in my chicken stock. Homemade chicken stock, though, is one of those things that can be surprisingly minimal. I've made it with literally a chicken carcass and peppercorns and it was still better than a can. If I have some garlic, onion, parsley stems, leeks, shallots, carrots, parsnips, thyme...I may throw some in, but nothing is really set in stone.

Yes. I actually often skip it.