I've mentioned this before, that AM radios are low tech, and haven't been really made for race cars in a while. Toys, yes, but not mass market competition level cars. Which means that the radios, receivers and crystals you find will probably be OLD.
If you are making a car as a runner, screw period-correctness and put in the good stuff. You can buy Spektrum DSM and other VERY good quality radios for less than that radio sold for in 1983, and the performance of the new stuff is infinitely superior to anything vintage—range, battery life, reception, adjustments. Everything basically.
In my opinion, R/C radio stuff has the least use in the vintage market because it isn't going to be used very easily, and short of the quick photo-op, it has little purpose for a collector.
Clean it with some WD-40 and wipe it down with some denatured alcohol, brush the dust out of the nooks & crannies, then stick it on a shelf behind your favorite RC10 queen, and you have a wonderful topic starter.
I think you'll save yourself a lot of headaches in the long run, but that's just me.
