If I were to be accepted as a transfer for fall 2016, could I do a summer session at the UC I was accepted at? Or can I not start until fall?
YES!
You can do a summer session anywhere. I just did one at UCB as a CCC student. But yes.
(well, not technically anywhere, but summer sessions are available at all of the UC’s to my knowledge)
I’m not sure if UCLA allows CCC students to do summer school. I might be losing my mind, but I seem to recall reading that.
Welp!
To answer OP’s question, if you are admitted for fall 2016 at your school of choice, then you can definitely do summer sessions there for summer 2016!
I’m fairly certain you can do summer sessions at any UC regardless of where you attend during the regular academic year. When I was at UCI I used to get emails from other UC’s saying “come do summer sessions at our campus!” The ones from UCLA used to make me so sad LOL. My friend from UCI is doing summer sessions there right now and she said she had some kids from UCLA in her math class, which she claimed threw the curve for the whole class lol.
Ah, I thought OP was asking to take UCLA classes while still a student at a CCC, which is what goldencub is doing right now. I don’t think you can do that at UCLA, but you can after acceptance. My bad.
Yes you can at UCLA. Any college student, CC student or high school student, as long as you meet any prereqs.
http://www.summer.ucla.edu/academiccourses
You can do summer sessions at any UC (even if you’re at a CCC, or another non-UC college), but you’ll have to pay the non-UC price per unit, which can be pretty hefty.
Berkeley is open to anyone during summer as well. There are a lot of domestic and international visitors during the summer.
Additionally if your admitted for the following Fall semester, they actually have the “Transfer Edge” program that allows you to take courses at the cheaper UC-student rate.
Source: http://summer.berkeley.edu/special-programs/transfer
I’m taking summer classes at UCLA after being accepted in April, so yes, you can.
Wow, thx @SDGoldenBear in my notes I have UCLA summer session currently not available to students outside ucla. Why do I have that? Obviously wrong. Hmmm, I guess I will never know. 
Thanks everyone. That’s so helpful they allow that. This will help me tremendously.
Just remember, you cannot use it for the 60 unit requirement.
I’m already there. I’ll be doing it to get a head start on the physics series.