If your friend's mom is your legal guardian or your court appointed guardian then the answer is "yes" she can enroll you in school. Even if that's not the case, you still have the right to go to school. It sounds like you need some support. My suggestion is that you ask your friend's mom if she will come with you to meet with the principal of the school you'd like to enroll in. I hope this helps and I wish you well!
I'm sorry to hear that you're having some difficulties with your mom, and no longer living with her. It's important that personal or family issues not prevent you from pursuing and achieving your educational goals.
However, in the United States, it is typically required that a parent or legal guardian enroll a child under the age of 18 in school. Unless you are a legally emancipated minor (and therefore have the legal rights to enroll yourself in school), please talk with your mother, father, or another adult family member about your need for their help with school enrollment. Please reach out to trustworthy adults in your life offline to get the support you need as soon as possible.
You and your parents (or adult guardians) may also contact Boys Town National Hotline, which specializes in helping teen girls and boys (and their families) with the issues that they face: 1-800-448-3000.
Thank you for asking, and please do not give up on getting your education!