To receive Amateur Radio plates, a copy of the operator's FCC Amateur Radio Operator's License must accompany the request, and the requestor must state that it is current. Upon revocation, abandonment, or transfer of the license, the plates must immediately be surrendered to the DMV or the transferee.

Notice that some of the plates are of the personalized series. This supports my theory that Amateur Radio plates are just a special form of personalized plate, but I haven't asked the DMV. This may also support John Carty's theory that applications for personalized plates are not checked very closely for call signs, and that they are not supposed to be on the caribou or mountain base.

Earl Jenson has quite a few of the older ham plates:

From AS 28.10.181:

(i) Amateur mobile radio station vehicles. A validly licensed amateur radio operator who presents satisfactory proof that the owner holds an unexpired Federal Communications Commission amateur radio operator's license of any renewable class, and who presents satisfactory proof that the vehicle contains or carries an amateur radio transmitter and receiving unit of a type applicable to the license class applied for, and who is permitted by law to operate a fixed station, may register one amateur mobile radio station vehicle for each radio license issued by the federal government and may receive for the vehicle distinctive registration plates instead of regular registration plates. The number on the plates must be the radio call sign of the owner.

The Alaska QRP Club
The South Central Amateur Radio Club