I LOVE the idea of designating tax return money for furbutt emergencies, but I'd be hesitant to give it straight to a regular vet office. What if the pup needs to go to an emergency clinic, a specialist, or a different vet (like if you happened to be out of town)? I think that is a more likely scenario than not being able to access funds in a bank. Maybe I'm biased because I live in one of the biggest banking cities, but there are ATMs everywhere, even when banks are closed - and I'd gladly pay any fee to access my money through another bank's system. Also - if you have the ATM card, you could just use that straight away at the vet office - again, I'd gladly pay any overdraft fees.
Levi's money is in a separate account. I can transfer the money to my general accounts at any time. This plan works well for us - I wouldn't be tempted to touch it on a normal day - but I can access it whenever needed, and can use it at any vet office or clinic we go to.