Remove the vinyl with a floor scraper. Smooth the floor with a belt sander. This will take off most of the remaining adhesive. Fill any low spots.
If you're putting in tile, the thinset will fill low spots just fine.
I'd pull up the underlayment and vinyl together, down to the original subfloor, before I would belt-sand a kitchen full of hardened adhesive with who-knows-what in it, and dispersing it through the house as fine dust. Of course, doing that properly would also mean pulling out the base cabinets. If the kitchen redo is a gut job, no big deal, but if only the vinyl needs a refresh, and the difference in floor height isn't a show-stopper, a layer of whatever they sell in your area for overlay underlayment (and around here, it IS luan, with printed screw marks on it) is the cost-effective way to go. Back in the stone age, we used 1/2" or 5/8" particle board as kitchen underlayment, but if it ever got wet, it was a major pita. Thin plywood, methinks, would stand the occasional oopsie leaking through a seam much better.
-- aem sends...
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