This is the first question you need to address when starting a vet fund: How much should I be saving up? The answer varies, depending on how old your animals are, how many you have, what kind are they [cavies or chow chows? rabbits or rhodesian ridgebacks?], and other questions. This is definitely something that you need to sit down and brainstorm on your own, but I will give you the guidelines I came up with way back before I adopted my bunnies.
An emergency could be a one-time visit, a couple hundred dollars and everything's fixed, or it could be a long drawn-out process. You really should plan for either. I recommend at least $500-700 per guinea pig, depending on your average vet bill/ER bill. To give you an example, the vet I see regularly is $45 per visit, but my emergency vet (Cornell) is nearly $200 for a fairly small emergency.
For rabbits, it's a good idea to have at least $700-900 on hand The reason that number is slightly higher is because folks are more likely to have 2+ guinea pigs, compared to one or two rabbits [not true for everyone, of course]. But if you have three guinea pigs, and one gets very sick, you have at least $1500 in care funds for it. Whereas, if you have one rabbit and it gets very sick, you only have $900 in care funds.
There is no set amount, no limit, to how much you should have saved, but the numbers I've given should cover you for basic emergencies. In five years keeping rabbits and guinea pigs I've only had to go to the ER vet once for Benner [Newman's brother] about a week before his death--he wasn't eating and his regular vet was on vacation--the only other emergencies I've had was Sophie-Mo and Olive's emergency spays [$240 each], and Newman's upper respiratory infection, which went downhill so fast after I noticed it that we couldn't get him to a vet in time.
It's best to have a vet fund before you adopt your friends, since almost no vet these days will accept a payment plan. [many do take Care Credit though, which is worth looking into if you have a lot of animals and no vet fund]. Some vets will work out a payment plan if you have a particularly hefty bill and you've been a client for a certain amount of time. My vet allows this for me, and I usually have my balance paid off within a month. I do this so that I don't have to take money out of our emergency fund if I don't need to.
It's also very important that you register with a vet before an emergency arises. Our emergency vet requires a $60 registration fee for each animal, so for my 9, that's $540. Do I have all of them registered? No. Three of my animals are registered there though, one of whom is still living; and the vet knows me from the House Rabbit Society. Some clinics, however, provide emergency care only to registered clients, whose animals have already had one routine check-up with them. Our regular vet clinic is one of these clinics, but our exotics vet doesn't do emergency care regularly, so we have to have a back-up.
The moral of the story being, have a vet fund and a vet before you need one.