I have a really low-grade laptop and I find that I run out of memory for rendering even very basic stuff. I have a thumbdrive and I was wondering I could use it in place of RAM. Does blender allow for renders to run based on storage and, if so, how?
This is not possible. A USB drive is a storage medium, much like your hard drive.
The only possible useful way of using it, would be adding or moving the operating system's swap (pagefile.sys on Windows-based computers) to this drive but even if Blender would make use of it directly, it would be very slow.
Unfortunately there is no cheap way to replace RAM with something else.