I am 15 and don't get along with my guardians who are my grandparents. I have many problems with them. At this age, is there any way at all that the courts will give me legal permission to live with a close family friend?

Who is the close family friend? Are they related to you at all?
Will your guardian sign for you.

To move out otherwise would require you get emancipated. You would have to have a job, enough money to live on your own, ( not with someone) You would have to pay all your own bills.

Also you have to have a valid reasons, not just having problems or not happy.

Why are you not with your parents ?

ANY question on law needs to include your general locale as laws vary by area.

But the likelihood is that no you will not be allowed to move out, even if your grandparents agree.