Welcome to PC--I can relate completely!
I wasn't physically able to sleep over ANYWHERE until I was 18, and I went off to college.
I struggled with sleepovers all my childhood and adolescence, and I would get so nervous I would throw up, too. What my big fear was that I wouldn't be able to fall asleep, and I would be the only one awake, and something bad would happen.
I still struggle with this, but it is much more mild. I have panic disorder and generalized anxiety disorder, and I found my miracle med when I was 16, that helps with both. But I still was only able to overcome it by going away to school. College is not a fix--I think my overcoming the fear was all about finding the right combination of meds and a good therapist, and understanding my fear and anxiety more.
If you can, talk to your mom and a doctor about it. They will be able to address the situation properly, and get you the right kind of help