Does your home university offer any sort of summer research program? I know a lot of places have formal programs funded by the university which would provide funding for participants. Even if such a program doesn't exist, you can always ask professors you know if they have a project you could work on over the summer. It is less likely that they will have funding, but it is possible that they would have something you could work on.
Doing an internship with a company would not be the worst thing ever, and certainly better than doing nothing. You would learn practical skills, and having that on your resume will be important if you decide not to go to grad school.
You could also work through a textbook or topic on your own, however that would require a good amount of self motivation to keep on track.
If you are really set on doing a REU I'd suggest applying to more programs, since it is fairly easy to do so. They are almost all on mathprograms.org, and don't have application fees. That means applying to more programs only requires writing another personal statement. Since your letter writers just upload a generic letter, applying to a lot of programs won't be a burden on them either.
Also, if you are only applying to pure programs, maybe consider some of the applied ones. I was hoping for a pure program, but I ended up doing an applied one. I really enjoyed it, and am now applying to PhD programs in computational applied math!