Vibrato tube going bad, either the oscillator section or the driver section; swap it and see.
Neon/LDR module going bad (Some Fenders)
Broken/open resistor, capacitor, or wiring
Faulty speed or intensity control
Faulty vibrato footswitch , jack, or wiring
Fiberboard contamination: Dust, dirt, and junk can let the LFO signal leak into the audio path. Vacuum the dust and dirt away, and if it still persists, remelt the wax top and bottom with a hair dryer.
Solder blobs from eyelets touching insulating board: Sometimes excess solder drips out the bottom of an eyelet and can intermittetly contact the insulating board, can cause ticking. Remelt the eyelets and examine the board underneath for any blobs dripped down.
Funny ground on some SF Fenders; On one of the signal tubes, the cathode cap was placed on the tube socket, and wired to a ground lug on the vibrato cancel jack instead of across the resistor on the fiberboard. The vibrato shares this ground line, and can the vibrato current can cause audible ticking in
the audio path. Rewire the cap to another ground or relocate it to the board.
Poor Signal wire layout: signal wires run too close to vibrato leads can pick up the LFO signal. Move them around and see if the ticking goes away.
Bad repair/replacement foot switch cable: the Fender footswitch cable is not two conductor; it's single conductor shielded, plus single conductor. The reverb wire is shielded, vibrato wire is not. This keeps vibrato out of reverb. If you retrofit with two conductor shielded, you get vibrato ticking onto reverb audio.
Sharp tick in vibrato oscillator: On neon/LDR Fenders, on the neon bulb side of the module there is a 10M to one side of bulb, 100K to the bulb; from the 10M straight across the board is the gnd point of the LDR. Put a 0.02 cap from 10M/bulb to the ground point; this works by filtering the output of the oscillator.