Nope, a medical card is a federal document that can be issued in any state. My driver's license is in Wisconsin and I recently obtained a new federal medical card in North Dakota.
Medical cards and not state issued. Never have been to the best of my knowledge. You see an Examiner in any state and they give you a medical card. They also transmit that medical information to the FMCSA and the FMCSA places the medical information in the CDLIS where the states can see it. Just get your physical and follow the same state that issued your CDL rules for reporting physicals. Your done!
My MEC expired yesterday. I've been on the road for almost 7 weeks so I couldn't get it done at home. Two weeks ago I stopped at the TA near Gary IN. There was a "clinic" in the lot, so I stopped in and got it done for $113. I then took pics of the papers and emailed them to the DMV in Nevada. Three days later the clerk at the DMV emailed me back and told me everything was updated. But I was really starting to sweat because I didn't think I could get it done on time.
I got mine the very last day at Petro Atlanta. Uploaded it to Michigan. Only get a one year so I try to stretch it out as much as possible.
The Feds don't care which state your CDL is from or which state your DOT card is from. Companies have their own view on the matter. Your company can have stricter rules than the Feds.