I'd avoid vegetable oils as they can get fungal growth in a damp environment and they don't contain any of the drier additives you need to ensure that the oil dries (unless of course you add terebenzine driers like those used by professional decorators). Instead use either a worktop oil such as those sold by Liberon, Junkers, etc (and they are proper professional products and in no way second rate) or alternatively go for a traditional oil finish such as polymerised tung oil or boiled linseed oil (although check the label for suitability).
To remove a lacquer or varnish finish first, though, I'd recommend scraping with a paint scraper such as a Skarsten hook scraper or similar (available from trade decorator stores such as McDougall-Rose or Dulux Trade Centres), cleaning up with a cabinet scraper then and only then sanding to a finish (grade P120). Scraping will remove the heaviest coating much more effectively and probably a lot faster than sanding can and is the way a cabinet restorer decorator would approach the problem to start with. To remove a finish by sanding alone you need something like the green or yellow aluminium oxide (alox) paper sold for decorators - P80 grit will probably remove the stubborn stuff with P120 to finish always remembering to sand in the same direction as the grain, but remember to wrap your abrasive paper round a piece of wood, or even buy a sanding block, as it will save your hands a lot. You will need to do a certain amount of hand work in any case as any machine will not be able to get into corners, etc.
Once the surfaces have been cleared of original finish wipe over with a clean cloth dampened with something like white spirits to pick-up any dust and leave to dry. Apply the oil generously using a clean, well-washed cotton rag cloth (again decorator stores sell bags of the stuff), rubbing in then wiping off any excess before it starts to become tacky (i.e. within 10 to 15 minutes of the original application). You'll need 3 or 4 coats like this, a day apart to start with, then one coat a week for a couple of weeks, finally one coat a month for a couple of months. Thereafter every 6 months. I'd recommend using a Scotch grey or maroon abrasive cloth before periodic oiling as this will smooth the surface out nicely
Note: cloths used to apply oil finishes need to be opened out flat and left to dry outside after use. This is because oils with driers added (most commercial oil finishes) dry exothermically, i.e. they heat up whilst dying out. so if you screw them into a ball and throw them into a bin you are risking a fire. Really!
Scrit