When do universities in the states have their summer vacation?

It depends on the school. You should be able to see when spring semester ends, along with various semesters
offered in the summer by going to their website however. If unable to find that, look for a phone number and call.

May to August, or June to September.

Most universities in the US let out in early to mid May. They typically return for the fall term in late August or early September. Some universities hold summer terms which (depending on the school) might draw a lot of students or only a few.

My kid's a professor. The last week of classes is next week. After that, there's a week of final exams, then the students are done until early September. My kid still has grading to do after that, and IIRC they allow five days beyond the end of final exams.

It varies from one college or university to the other, even ones in the same system like the SUNY schools. If you want to know about a specific one, visit their website and view its academic calendar.

From the end of May to the middle of August.

It depends on the school. For me, It's from May to August. For other schools, it may be June to September.

Ours finished exams last week. We'll go back the last week in August. There's no standardization of university calendars in the US.

May or June through August or September

Mid to late May to late August.
My daughter was done by the second week of May

Check the web?