I'd say you'd be hard pressed to find an internship (especially for pre-vets/undergrads) that is paid. I've perused many an animal internship and can't think of one off the top of my head that is paid. A few will provide at least a small stipend for food, or allow on-site housing, but I'd say those are in the minority.
I'll double check and see if I can't find one of my "saved" internships that fits your criteria.
EDIT: So, I have some time to kill, and was googling for more equine related things as most of the internships that I've looked into have been more exotic/aquatic/wildlife and I came across this site:
http://equinescholars.georgetowncollege.edu/opportunities.htm#internships
Now, I know it's specific to that particular college program, but you might have some luck contacting the clinics directly and seeing if you could still do an internship there.
A lot of other equine related clinics etc. seem to have internships exclusive to just-graduated veterinarians, but similarly, you may have some luck just trying to contact different clinics and see what opportunities may arise.
OR aim for something more part-time, and get a job on the side.
Sorry I'm not more useful, trust me I understand the struggles of trying to find internships but not go broke! Good luck.