see title
Do universities or high schools usually let students take it?
The school has to register as a test center. Technically, any school can do it, but I know quite a few applicants whose schools wouldn’t do it, so they had to travel long distances to get to a test center. It really depends on whether the admin are willing to put the effort in. There’s a test center in NYC, though, which is one of the cities you were interested in.