It really depends on how long you are looking for her to stay.
I'm kind of in the same situation as you trying to get family over here myself and the Covid-19 pandemic sure has changed some policies or has made them confusing, such as the 30-day tourist visa for U.S. citizens upon landing.
As it stands with her being a U.S. citizen, the best route that she could take is to get a tourist visa which would allow her to stay in the country for up to 90 days. This would need to be applied for at the nearest Japanese Consulate or Embassy to her.
During her stay, a visit to
Naha District Immigration Office (or
Kadena Immigration Branch Office) would allow her to apply for a 6-month tourist visa extension (see:
more requirements). These are generally granted fairly easily if a tourist can show means to live which could be something along the lines of living with you rent-free to a bank account with enough money for a flight out of here.
Though, you do mention that you are SOFA status and don't mention her age and get into the finer details of that, for obvious OPSEC or other privacy reasons. You could get with your chain of command to see if they would temporarily grant her SOFA status. Most likely, it won't be accommodated, but it's worth a shot.
You also don't mention whether this is a medical emergency for you or a medical emergency for her. I'm just assuming with this statement here that the latter could make obtaining a tourist visa or extension impossible because the Government of Japan could determine that they don't want to pay if it's an obvious emergency that they'll need to assist with.
If you could share any other details, I or someone else could possibly help you narrow down what the best route is for you to take.