The bit you really need to lubricate is where the shaft of the tuning handle runs through the "saddle" especially on the handle side. This is where most of the pressure is put. It would be nice if you could put a thrust washer right at the end, but its rarely possible without removing the handle. The cog gear and worm don't actually need much lube at all; there is supposed to be resistance here, that's how a worm and gear works. in fact if you lube it too much, the gear can slip, and I have video to prove it!
I use light machine oil, WD40, graphite ... whatever i can get in there. I don't use vaseline, too sticky IMO and collects dust.
I use a soap paste to lube where the shafts of the tuners run through the pegbox. It dries, but stays slippery.