How do I get a summer research position as an undergraduate? All the questions here on this topic seem to be about masters/PhD students, so I'm not sure if that applies to me. I am an international student studying in the US, which means the only place I can get a full-time paid summer job without mountains of paperwork is on campus. I would love to get involved with a professor's research over the summer, for valuable skills and experience. How do I go about developing that kind of rapport/relationship with a professor? It feels weird to just do a cold email saying "hello, I'm really interested in your research, do you have anything going on over the summer that I could get involved in?".
This is a tough situation to be in, as a lot of (STEM) undergrad research in the U.S. is funded by the NSF through REUs, and only U.S. citizens or permanent residents are eligible for those.

Many universities in the U.S. have school-wide summer research programs for their undergrad students, with various sources of funding (that you may be eligible for). Usually you apply directly to the program, instead of approaching individual professors who participate. Check if your school has one. You can also check if your school has an agreement with another school to "share" undergrad summer research students in such a program. (My university has such an arrangement with a few schools in the area, in which their students can participate in our school-wide undergrad summer research program.)

Ask your academic advisor (i.e. the person who helps you decide which classes to register for) if your school or department has an undergrad summer research program. Also ask your academic advisor if he or she knows of any faculty members in your department who are looking for an undergrad summer research student, outside of any official program. (That's how I got my first undergrad summer research job!)

Besides that, you can definitely approach professors in your department individually. My lab seriously considers every undergrad student in our department who reaches out in this way. It's helpful if you mention what your research interests are, what classes you have taken that are relevant to the research we do, and any other skills you have that might be helpful for our research. (You should also mention that you're an international student, since as I mentioned, we may have to work out funding in a different way than we usually do.) If you are taking courses with professors whose research is interesting to you, you can go to their office hours and ask them about it there.