It is in their power to prevent to from taking it at your HS, though. They are under no obligation to order you a test. But they would be hard pressed to prevent you from taking the exam at another test center.
What math courses does your HS offer post-Calc AB? Personally, I think you are better off following the math curriculum at your HS, and looking for community college options once exhausted.If you’re hell bent on self studying calc, look through past threads as the question gets asked every year.
In terms of psych, while not a strategy I’d recommend, having never taken a psych class, I self studied it over a weekend with Barron’s (which is sorta why colleges don’t put much stock in simply self studying for an exam).
FYI, the fact that the AP Psych exam is so mind-numbingly easy is one of the main reasons Dartmouth eliminated AP Psych credit.
http://pbs.dartmouth.edu/undergraduate/curriculum/transfer-and-ap-credit