Can I get custody of my grandmother without her last name and without proof she my grandmother?

Jessie has given you a good answer about guardianship. If you want, you could explain the situation and what you hope to accomplish by getting "custody," and maybe some posters will have other suggestions for meeting your objectives.

By custody I assume you mean guardianship. The probate court grants guardianship. It is not a simple process. What you need to do is consult with a lawyer to petition the court for guardianship. This involves having doctors declare her incompetent and a thorough investigation of your finances and background, particularly if there is no proof that you are related. If she does need a guardian, chances are the judge will appoint you as guardian or appoint someone else. This is not something to enter into lightly. It is expensive and you will have to make reports to the probate court about money spent and actions taken. Good luck.