You could go ahead and start your training locally in whatever is available. You just won’t be able to solo without the medical. Then when you’ve done all you can do locally and without soloing, make the transition to LSA and finish up.
Good luck.
I would suggest mixing this with an accelerated course, if you cannot find something local.
Obtaining a certificate requires 3 basic educational accomplishments:
1) Hours checklist as defined in Part 61 - This includes dual and solo minimum times, and what needs to be covered in those times.
2) CFI signoff authorizing you to take the formal exams
3) Formal exams - written, oral and check ride
The last 2 can't really be done without an LSA, because you cannot solo in anything else without a medical, but most dual time can be done in practically anything fixed wing single engine. In your position, I would suggest starting with the dual training with what is available. While you complete your initial training, try to get involved with EAA, and AOPA gatherings in your area, tell them what you are trying to do. After you have completed ground school, the written, "checklist hours" dual, and feel you are ready you can then move forward in one of 4 ways.
1) You may get lucky and find someone who owns an LSA that they are willing to rent for you to finish up. (Not likely, but not impossible)
2) Find and purchase an LSA
3) Commute to an airport that has rentals
4) Find an accelerated program that offers custom "finish up" training, you may be able to finish in a few days or schedule lessons around caring for your wife.
