I have a 1965 Thunderbird and recently the off-on switch no longer “clicks” signally it’s powered up.
Replace the radio. No sense trying to fix this one. A new one will cost less than getting this one fixed.
To replace the switch, it needs to be replaced with a switch of the exact size and shape in most cases. Tracking down the right switch can be time consuming and fruitless, if this radio is as old as the car. Then, the old one needs to be soldered off and the new one soldered on. If it is a more modern radio, the switch is probably very small, and hard to replace (extra labor time).
If the off-on switch is combined with the volume control, the switch is ganged with the volume control and rides piggy back to it. The volume control section is in front and has three connection points while the switch is in back and has two connection points. One way out would be to bridge the two switch connections. The radio will always be on. Then put a switch in the power line to the radio and mount the switch in some convenient place under the dashboard. This switch will cost about $1. The car won’t be completely original, but this is a cheap way out. I did the same thing with a household stereo receiver–bridged the internal switch and put an inline switch on the cord. It worked just fine.