I did quick and dirty maintenance on a Gemini I G-12 to get it safe and working... 3-wire cord, power supply capacitors, whichever other caps I could replace.
When I was "done", it played fine but... the vibrato still didn't work!
Don't let that discourage you... I didn't replace caps in the vibrato circuit because I didn't have the right value (smaller than coupling caps), and I bet that would have helped.
Why did I replace others and why am I so suspicious of the remaining ones? Not sure what type the original caps were, I think paper-in-oil (electrolytic in the power supply of course). All were discolored, some misshapen, some appeared blackened from heat. The coupling caps could leak DC and cause bias problems leading to major sound issues or bias leading to tube failure so I considered replacing most caps to be a safety/reliability priority... but they also looked terrible and I believe they're a type known to wear out after 50+ years.
So, be ready to replace almost every cap, especially in the problem area (vibrato circuit)... if that maintenance hasn't been done before.
Deoxit (Home Depot sells generic electrical contact cleaner if that's easier to get) should improve pots. Like caps, they need to
work and be roughly the right value for the vibrato to do its thing. This also improves audible scratchiness or cutting out heard with audio pots.
Resistors (other than pots) "can" go out-of-spec over time but I feel they're unlikely to be
so bad the amp, or a feature like vibrato, no longer works.
This is the same advice I'd give for any problem in an old amp: safety stuff if needed (3-wire cord), capacitors, clean pots. Resistors and bad tubes are a possibility but IMO towards the bottom of the list.
PRR gives other good suggestions with less drastic solutions.
My recommendation would depend on what other work has or hasn't been done on the amp in the past but if basically nothing has been replaced since the 60s, a lot of "preventative maintenance" is due or overdue regardless of which problem is actually happening right now.