Can a private pilot receive recurring training while waiting for a special-issuance medical approval? In the last year, I have had to undergo bypass surgery. I am fully recovered now (and past the mandatory 6-month waiting period), but have just been apprised by my AME that my 3rd class medical deferral is likely to take a month or two for the FAA to process.

I am also in a partnership, and our insurance requires annual recurring training. Normally, all four of us (partners) take the recurring training at the same time, and get a discounted rate as a result.

As fate would have it, this year's recurring training will fall into the time I am waiting to hear back from the FAA. Since the flight portion of the recurring training (essentially an IPC) is conducted under the supervision of a CFII, can I complete it without a current medical certificate?
My understanding is that your medical is currently invalid (because you had bypass surgery and need to get your medical reinstated), so you currently cannot exercise the privileges of your certificate - i.e. you can't act as pilot in command of a flight (or any other required crew member in a role for which a medical is required).

As Casey pointed out, you've got an out that allows you to still take training (and even to fly generally): Someone else with a valid medical and a certificate appropriate to the category &amp; class of aircraft needs to act as the (legal, FAR 91.3) pilot in command while you're operating the controls.

If your CFII is willing to accept that role (and the attendant liability) while giving you your training you can go up with them and complete the recurrent training your insurance company requires, but this is something you need to discuss with your CFI prior to going up with them.