I recently had a seizure and have been told that state law prohibits me from driving for 3 months. Can my unit force me to use Uber?

What WILL happen is you will be required to be at your appointed place of duty at the appointed time.
If you can not do that from an off post location, they will likely find you space in the barracks and then tell that Staff Sergeant NCO, figure it out, walk, bum a ride, take on post transportation.

If they are smart they will not tell you HOW to accomplish your task, just remind you of what the tasks are.
If you prefer to live off post, knowing you don't have personal transportation for the next 3 months, its up to you..how are YOU going to work it out? That is your responsibility.

I don't see how they can FORCE you to use Uber. How you get to work is your responsibility. If you are late too many times, they can tell you to move into the barracks.

No, your unit can not force you to use Uber, however they can force you to follow the law if your driving right were suspended. As far as getting to work, it is still your responsibility to report to work and to make necessary arrangements to get there. In saying this, you would need to make arrangements with your section NCO's to assist you during your time of need. That's of course it they are good leaders.