It's irrelevant what "most" parents do or what's "normal"; you're in a specific situation where you (I think) told your daughter that you would pay for her lessons and paperwork to get her license and now you may not be able to follow through. Likely she feels blindsided; at 18 just out of college working full time to save for starting uni, she's going to be counting and accounting for every penny. (Keep in mind, she doesn't know what uni is like or what it will cost her or how she'll manage; this is all new and still speculative to her). Just as the large unexpected vet bill was a blow to you, the bill for leaning to drive will be to her. Of course, she shouldn't speak to you as she did, but she's not "just" being an entitled brat (about this issue, anyway). She may not have fully absorbed the issues with your pet or what that expense means in your larger financial situation. If you really cannot pay, tell her why.
Otherwise, I'd take another look at your finances and decide what's reasonably possible now. Is this mainly a cash flow issue - e.g., driving lessons was going to absorb a particular pool of discretionary money for the next few months, and now the vet bill has hit at the same time? If so, could it be an option to have your daughter pay up front (or split the cost) and you "pay her back" later on/incrementally?
As a side note, it does sound like you're not comfortable with how things are working out over this gap year. If she's spending her earnings from her current job frivolously/buying things you consider too much of an indulgence, I can see why you're annoyed. If she's mainly using the temporary opportunity of full-time income and very limited expenses to save all she can for uni, I'd respect that but again, be realistic. If you need her to pay for certain things, or feel it's important for her that she does, arrange it up front before she's mentally added her earnings to her "uni savings" total. And, since you mentioned it, she should be doing a share around the house - at the very, very least taking care of her own needs, keeping her room clean, and tidying up after herself in common areas. But decide what needs to change and discuss it with her, no (more) surprises!