As a Vandy '08 grade myself, I know how hard it can be to find what you're looking for. A 1 bedroom stand alone home is going to be incredibly tough to find in that area, especially for the price you're looking for. I do see one duplex that is available on Natchez Trace for $895 a month, but it does not list if pets are allowed or not. Is there anyone else in your program that you might be able to room with? There are quite a few 2-4 bedroom rentals in the area.
As you can probably guess, the large majority of rental units in Nashville are simply posted on Craigslist because they do not want to have to pay an agen to market their unit. I built a website that does allow you to search all rentals that are currently on the market that also have an agent. Just go to this site and click rental on the "property types" tab and type in your criteria. Again, these are only the units that use agents and post their rentals on MLS.
I would also ask your housing department about their online listings. I do know that Vanderbilt has a website that lists rentals in the area that are specifically advertised to Vanderbilt students. Lastly, just drive around the Vanderbilt area, especially the Blakemore and Natchez parts just to see if there are any for rent signs just sitting in the yard.
If you would like any further help, or would like a tour around the area, feel free to give me a call. My name is Scott Pettus and all of my info is on that website on the bottom. Have a great rest of the week.