How can I legally manage my mother's funds while she's in a coma?
Your question implies that your mother did not have a Durable Power of Attorney which is a document that would have easily and without court involvement provided you with the authority to act on your mother's behalf in regard to her financial affairs.
Therefore your only option is to go to court and be appointed as your mother's guardian so that you can make both health decisions on her behalf and manage her finances. In a situation, such as yours, the law provides for you to be appointed quickly on a temporary basis while your petition to be appointed on a more long-term basis is pending. This is not something that you can do by yourself. I strongly urge you to contact a lawyer to help you with this.