You could apply for Pell Grants which might be able to help with your living expenses besides college if you want to go to college. I'm not really sure how it works though. Otherwise, if you can get a job, get a job. Don't apply for SSI unless you really cannot work at all. It should be like a last resort. The Website for SSI is www.ssa.gov.
You would not be entitled to benefits as a dependent grandchild. One of the requirements is that both of your parents must be either deceased or receiving disability benefits from the Social Security Administration.
If you've graduated from high school I suggest that the time has come for you to get a job and see about lining up a roommate or two. Taking control of your own life might alleviate your depression.
Because you haven't worked you wouldn't be eligible for social security disability. If you want to file for SSI (Supplemental Security Income) you can call 1-800-772-1213 and make an appointment to file either in the office or over the phone. However, since your depression hasn't been severe enough for you to receive treatment for it, it is unlikely that your claim will be approved. Depression must be severely disabling in SPITE of treatment.
I am sorry about your grandmother.