I spent four years working for a similar (local) type of pet assistance organization.
One HUGE mistake that people make is just contacting the most-local org and not going any further. All of these orgs operate on donations, and the need always exceeds what is available.
Do everything you can to find MULTIPLE orgs that might be able to help. If you hit a single org with a need for $1,500, they will often refuse. But if you tell them that you're trying to contact as many orgs as you can to spread out the loss, ask them if they would go along with that -- most will. If a $1,500 loss is divided among six orgs, that would be $250 apiece, which might be more do-able.
The most-local org may want you to take your pet to another local veterinarian that gives them a financial break.
Be sure to have all of the necessary info right in front of you: type of damage, type of repair needed, vet clinic name/address/phone, specific vet's name, estimate of repair, etc.
And, I'm not saying it would necessarily work, but ask the vet in charge if you could contribute labor in place of money, by doing things like cleaning kennels or pulling weeds, etc. If he/she happens to be short-handed, they may accept.
And, for anyone coming upon this thread that might try to exploit any assistance organization for their personal financial advantage: THEY DO NOT DEAL IN CASH. They send a faxed authorization of promised payment to the veternarian in charge. YOU get absolutely nothing. There's no "Just give it to me and I'll give it to them". No organization like these just fell off the turnip truck.
