Yes, you can register at community colleges just like anyone else. If you will be attending community college, be sure to check Welcome to ASSIST
to see what courses can transfer to a UC or CSU.
Among four year universities...
Berkeley specifically allows high school students to register during summer sessions. | Berkeley Summer Sessions
San Jose State does not say anything prohibiting high school students from registering during summer sessions. SJSU : Summer Session
Both Berkeley and San Jose State allow non-regular students to take courses on a space available basis during regular semesters: Concurrent Enrollment Overview — UC Berkeley Extension What is Open University? — Open University
You may want to check the policies at other universities in the area as well.
However, if you intend to transfer as a junior to a public university in California, the admission and other policies are generally more favorable to those whose course work was entirely at California community colleges. Some refuse to take transfer students who have too much college credit if any credit is from four year colleges or universities.