I picked up the small amp at a local music store - a Vox brand amPlug powered speaker cabinet. Unfortunately, I haven't quite gotten it to work for me at all. Power is fine to the instrument (I presume), but when I plug the line from the theremin to the 'guitar in' port, it does nothing regardless of what I tweak on the theremin dials. I am suspicious that the 'amplug in' port on top of the unit might be where the actual attached amp goes, and that what they sold me was JUST the speaker unit or something odd like that. I will have to puzzle over it and figure out where I went wrong. :)
I took a look online and It looks like the Vox amplug is just a speaker cabinet for a guitar headphone amp. Kinda like those little speakers for MP3 players, it is expecting a signal in that is high enough to drive headphones. Also with a 3-inch speaker it is likely to sound like a Radio Shack walkie-talkie. The mini amps are really cute novelties, but that is about all. You would be much better off finding some standard guitar practice amp with a much larger speaker.
There is a wide range of guitar amps that will work with the Burns. The best results will probably come with something more expensive and with a 10 or 12 inch speaker. Less expensive amps with 8 or 6 inch speakers may work OK but not sound the best. The toy amps just shouldn't exist, they are novelties like a rubber Richard Nixon mask.
If you are on the cheap, check Craigslist for a guitar practice amp. But be aware, a great theremin tone from a great amp will help to keep you enthused and working hard after you discover just how difficult the Theremin is. You may get by with a cheapie to start with, but be prepared to upgrade.