We just did it this summer. Had a California placard until it expired this year.
Procedure: South Dakota website has a form your doctor needs to fill out for you.
Ours went like this.... We were in Montana already when we found out we needed the Dr in California, to fill out the form.
Since we can communicate with our doctor thru email, I asked if we could email the form, have the dr fill it out and they mailed it to our mail service in Box Elder, SD.
We picked up the mail (since we were going to be there) with the form signed by our (his) dr, filled out Bob's part, he went in to SD treasure in Rapid City (where pick up license plates) and came out with the placard.
(FYI... The website tells you where to take the form and get the placard.)