The easiest option is staying in the dorms, through the summer housing program Vanderbilt offers (link: https://www.vanderbilt.edu/meetatvanderbilt/intern-housing/ ). The only negative is that it’s on the expensive side and the program doesn’t officially extend through the entire summer (although I think you could probably work something out for may since people stay in the dorms for maymester). It can be worth it though if you don’t mind the expense, and don’t want to deal with the headache of trying to live in an off-campus sublet.
For finding off campus sublets, a good option is the vanderbilt classifieds. You need a vunetID to see things/post so you know that it’s more likely the people you are leasing from will be trustworthy, and there will be stuff that is closer to campus. (link: https://www.vanderbilt.edu/classifieds/ ).
The other option is craigslist. Just use the search terms “vandy” and “vanderbilt” and you should be able to find something that is pretty inexpensive and close to campus, but it might take some time to find and you might have to deal with some annoying people.
I did research one summer, and I think I would probably recommend staying on campus if you think it’s affordable. I stayed in an off campus apartment, but trying to find a place that was cheap and close ended up being a huge pain, and if I did it again I might just save myself some trouble.