Dear Puzzled_Life,
It's a great idea to get a student loan and focus entirely on your studies. I highly recomend you to get your Social Security as soon as possible. Not only regards to laon but lets say even after you graduate, life would be really hard without social security number. From getting apartment, home, cellphone, cars etc or any kind of financing for that matter would be impossible. Remember it's never too late to start.
Ok now coming back to your intial question, Yes you can get a loan with a co-signer who is a citizen or permanent resident of the US. Here are few question that you need to focus on. Does your co-signer has a good credit score and has a long credit history? Specially these days bank are pretty hezitant to give loans, thus you want to make sure you not only get a cosigner but get a co-signer who has a good credit history and a good income. If you can do that then you can apply to lot of major banks who offer student loan. Dont just apply to one bank and wait to hear from them and if they dont give you good term then apply another one. You apply to as much banks as possible in a single day. That woldnt not hurt your co-signer credit score. But if you apply week after week then yeah it would.
I hope I have answered your question. Please let me know if you still have more question regarding this issue. I wish you all the best for your future studies.
Warm Wisher,
YekloYatri