In case you just want the look of a vintage radio (and don't mind that it's actually modern), here's what I did - it's a Crosley CR-82 AM/FM battery (4 AA) powered radio - they show up on ebay regularly for around $25-$40.
I feed a MP3 player (filled with period music) through an FM transmitter to it.
It's mounted to a wood tray I made with scraps that were laying around.
