What to do when a Soldier is consistently on quarters?

Sounds a lot like malingering to me. If this Soldier is an NCO your best bet is to counsel repeatedly and address it in their NCOER. The harder question is whether or not you believe this person should be separated from the Army or not. There's a chapter for repeated malingering. If they are a SSG, a negative NCOER will refer them to the QMP board for review.. The fact that she can't be relied upon to accomplish a task is enough to begin your counseling with. If these are legit medical emergencies it may also be time for the commander to look at asking the provider about a med board or a chapter 5-17

Can't drive a GOV due to meds but has been seen driving her own POV? If possible, contact her civilian provider and ask if the SM has been seen there recently. Obviously, you can't ask what for, but see if you can ask if they were in fact evaluated. If they are able to answer without violating HIPPA, and state that she has not been seen there, then you have her dead to rights for possibly the following:

Insubordinate Conduct toward a Non Commissioned Officer/ Article 91
Dereliction of Duty/ Article 92
False or Unauthorized Pass Offenses, False Pretenses/ Article 134

I would certainly press for the Fit For Duty evaluation.

Is this Soldier Active Duty? If they are, there is no reason for a civilian doctor to be signing off on quarters. She needs to go through PCM, even if PCM refers out. Even if Active Guard or Reserve, they should be going through PCM. Once, maybe even twice, I would let it fly, with the very clear instruction that in the future it would have to come from PCM.

If the PCM (who should be monitoring health records and be able to identify a pattern) thinks everything is legit, it is probably legit. If you STILL have questions, a quick conversation between the Commander and the PCM can clear things up and/or tip off the PCM to possible malingering.

Regardless, IMHO, this is really not your fight, it is the Commander's. Yes, you are the one on the front line, picking up the slack and seeing the problem first-hand, but this is one of those situations that is specifically on the Commander. If you have not already done so, you NEED to sit down with the Commander and discuss the way ahead.

Regarding the doing things on "off" time that she can't do during duty, I would recommend that you sit this Soldier down each and every time they get quarters and counsel them. Document on paper exactly what the Soldier can and cannot do. And then put it in writing that they are NOT to do any of those actions until date X or until re-evaluated by a medical professional. Have it come from the Commander, if possible to add extra weight. So if she cannot drive a GOV, then she cannot drive, period. Make it VERY hard on her to continue her shenanigans. One of three things will happen. 1) She has legit issues and will comply for the sake of her own health. 2) She is malingering, but will comply for the sake of her career. 3) She is trying to play the system and she will continue to say one thing and do another. Then, if nothing else, you have her dead to rights for disobeying lawful orders.

And if she says she can't come in because "quarters," call bull shit. VERY few doctors orders say that a Soldier can't come by to process paperwork. Literally every single time I have gotten quarters - to include the time I got it for infectious disease - I was allowed to stop by the office and drop off the quarters slip and check in with my NCO. And even if she, for some reason, CAN'T go to the office, you can certainly go to her house (or the doctor's office / hospital). After all - if she is so bad off she can't drive a GOV, it would only be PROPER for you to escort her home and make sure she is safe, right (bring a battle buddy / witness!)?