Aloha Jack,
As Mad Dog implied there are a several potential problem areas. If you have eliminated tires and wheels as a cause of the vibration then you can start looking for worn steering parts and front end suspension.
The steering mechanism, cam and peg follower and the bushings in the steering box wear particularly if not regularly lubricated. The peg that rides in the cam groove wears down and steering gets sloppy. There is an adjusting bolt to press the peg dowm but eventually you run out of adjustment.
Worn tie rod ends will cause play in the steering. The idler arm fulcrum pins are in a rubber bushing that perishes and also causes this play. Jack up the front end of the car and if you turn the steering wheel without wheel movement or up push one tire to the side and the other doesn't move, you have worn steering parts. By movement I mean more than 1/8 turn of the steering wheel and maybe a 1/2 inch of side to side movement of the wheels.
The front end suspension has several rubber bushings for the upper and lower wishbones. These wearout over time and make it near impossible to keep the front end in alignment. If you are going to replace these parts, I would recommend you use polyurethane bushings instead of rubber. These are available from most Triumph parts suppliers. They will not deform or perish like the rubber bits. While you have it apart, check the ball joints and shocks to see if they need to be replaced. If in doubt, I'd say replace them.
This may be more work than you were considering, but if this type of work has not been done in the last 10 years or so, it's probably past due.
Safety Fast,
Dave