How do I file for POA? I recently found out one of my siblings was stealing money from her.

Take your evidence and go to the police as well as APS - that's the action you CAN take now.

Too late for POA.
Guardianship is needed but it takes time and perhaps she does not have that? Since there is no DPOA just who is making the health care decisions for mom? How did she get placed on life support? I'm wondering if she has had an emergency ward of the state placement order done to get this done. If so, that person is in charge of mom and decisions for her person.

You may want to think about what life support could mean for her. Speak with social work @ the hospital. Does it look like it is short term (like placed in coma while swelling decreases) or is her condition probably permanent. If more long term, then Why life support and not hospice care? Long term life support will be very very very expensive (like what happened to Sunny VonBulow) and usually in a private facility so there could be no funds left when all said & done.

Mom cannot appoint any POA at this point, if she is on life support and unconscious. The only route now, is Guardianship. Also, you could notify the local sheriff and police as to the monetary theft, and possibly a "hold" could be put on mom's bank accounts, etc.

You will need to file for Guardianship since she is incoherant. Contact an elder law attorney to help you. Maybe they can get an order to freeze her accounts until that can be accomplished.

Sorry, your mom has to appoint a POA, it is part of the advanced planning we should all do, including writing a Will (I hope she has at least got that done).
How does your sib have access to your mom's accounts, and how do you know there is money missing?