How do I acquire POA for my adult son who has a brain injury?

If your sons brain damage is severely compromising his mental cognition, legally he can not grant POA authority to anyone.

What you would need to do instead is file for guardianship. If it is possible that his mental cognition could improve to the point where he could make decisions for himself - you could apply for “limited” or “temporary” guardianship.

I’m so sorry that this has happened to your son. You must be heartbroken and stressed beyond measure. Please try to take care of yourself!

I have an adult son with an anoxic brain injury due to lack of oxygen as an infant. I have POA on him, obtained when he became an adult. I gave him the forms and he willingly signed for it in front of a notary public. I doubt he fully understands it, but he trusts me and when I say something is needed he generally will go along with it. It’s not ideal but it works for us.
I’m sorry you’re in this situation. Your answer will depend on your sons level of cooperation and understanding. Best wishes to you both