Someone may have more fruitful suggestions, but if you don't get anything else, WWOOF is a possible option. I had several friends do it after Peace Corps to help extend their travel since they got free room and board. The premise is that you work on an organic farm, but several of my friends ended up (by chance or choice) working on a farm that also had animals which they did a lot of tending and care for.
The upsides are that it's fairly cost effective (you generally get room and board in return for your work so your major costs are the ticket there and your in-country travel and entertainment) and would be a very unique experience. The downsides are that you would probably have to put more work into finding an option that had animals and also was conducive to internet access for your work. You could likely get a letter from the owner of the farm, my friends that did WWOOF became close with the people there. The other minor downside is there would most likely not be a veterinarian at least for the WWOOF part so it would be animal experience and not vet.
My friends who did WWOOF also traveled a bit while they were there, so it's possible that you could expand and do other small projects/experiences that might have you in touch with a vet or would count as research. When I was in Peace Corps, I had the opportunity to work on some projects with local and international universities on fisheries and speciation (I left these small projects off of my vet school application which I'm really mad at myself about, but I will not forget them if I need to apply another cycle, haha). They were happy to have me help because I was in-country and free labor for them. The speciation project I did was with a team from a US university that was set up in-country - I did fish collection at my site and in return I got to work in their lab assisting with the PCR work on the fish collected around the country. They were working through a local university, utilizing their laboratory facilities so I got in touch with them through that university to learn about the project. There was also a university in the south part of the country that had a wildlife studies program and they did work on-site with rehabilitation. I reached out to them and was able to go observe at the school while I was in the area for a Peace Corps related training. Long story short, you might be able to do something like WWOOF and then reach out to a local university or other group to see if there's an opportunity for you to participate in other programs or lend a hand in short term projects that could diversify your experiences.
Another friend of mine got her dog spayed before bringing her back to the US. It was such a bizarre request in the eyes of the veterinarian there, they did not generally do those in her country of service and they ended up letting her watch the surgery. You might also be able to find a local veterinarian willing to let you shadow in-country on days off from WWOOF which I imagine would be a very unique experience for your app. The downside is that it might be difficult to coordinate from the US (or where you are now) and you might just have to take a chance on locating someone in-country once you are there.
http://wwoof.net/
Good luck!
