I'm going to address the Elephant in the room: the tiles. A vinyl product will show the tile grout lines. Vinyl is soft and flexible. It will ultimately sink into the grout lines causing 'telegraphing' (showing the tile pattern underneath). This is a HUGE issue with vinyl. The only way to stop that from happening is to sink the tile in cement (cover it in self leveling compound = SLC). What are your plans for flattening this floor (getting rid of the grout lines)?

Now to the question you are asking. A FLOATING floor is NEVER allowed underneath cabinetry. If the cabinets touch the ground (on boxes) then your floating vinyl floor will never run underneath. Only a high-end 'floating' cabinet install (mounted on the walls without legs) will have a floating floor installed up to the wall.

For jobs like this - SetaGrip is a great option. Www.setagrip.com has solutions for renovations like nothing else on the market.

Saw this is a few years old, but want to address the other elephant in the discussion. Installation of LVT is usually something along the lines of: floor prep -> lay down 5mm sureply -> lay down quality underlayment (1191177 is the current product number from Lowes, Soft and Sound 1.5mm underlayment as both a vapor barrier and insulation -> luxury vinyl tile. Unlike sheet vinyl (which SJ McCarthy's example plopped right on the tile would be correct), what I describe is vastly more robust and will not show existing substrate grout lines.

Doing wall-to-wall install should not be an issue if the opposite walls are left with a gap between the edge of the LVT and the walls proper. The room looks to be no bigger than maybe 14 x 20, so the cabinets would anchor only one end and not a large stretch. The wall gap would be covered by base and quarter-round.