Your problem could well be a badly out of balance tyre which is easily cured. lift the wheel off the ground and make sure brake shoes are completely clear of drum. mark the tyre with chalk in one spot and place the wheel in two or three different positions to see if it rotates to same position. Heavy section will be at the bottom, balance weight opposite to rectify. A bit of trial and error, or remove the wheel and get a specialist to balance. no movement would indicate not out of balance.
Vibrations from an out of balance wheel/tire does usually not go away with an increase in speed. I'd go with the suggested steering damper, but I know doodley squat about sidecars.
It's to do with the drag from the undriven sidecar wheel, as the bike accelerates, the sidecar pulls toward the kerb (due to friction on the wheel) and the rider tends to over steer in the opposite direction creating a wobble. Most European bikes up to the 60's had a friction damper in the middle of the forks and screwing down the damper reduces the tendency to oversteer and thus reduce the wobble. Once the machine has accelerated past this point and the sidecars foreward momentum matches the bike, the wobble stops......