Can you waterproof Luxury Vinyl Tile floors without grout?
In some areas, you can get away without having grout, but the tiles will be more susceptible to damage because grout (or something) adds strength to the floor (and keeps the tiles in place). You may have seen or heard about how some people make really tight fittings with beveled tile, and they don't use grout. This is more for areas that are onramental or where contact with water is unlikely.

In the bathroom, I think it's safe to say that you need something between the tiles especially because there will be water (on the floor). Another purpose for grout is to protect the underlayment from water... water damage could be summarized as swelling which can cause lifting/ungluing of tiles and warping and loosening of the wood and rotten/weakened wood or just plain mold and mildew. And yes, cleaning will be an issue without grout or something between the tiles.

There are alternatives to cement grout; you should search online for grout alternatives. Epoxy and caulk come to mind. If you don't put something in there, something else (dirt) will eventually fill in the gaps.

Edit- (based on the comment "my ex-husband already laid the tiles without spacing for grout")I don't like to promote products, but I'm not aware of any other silane vinyl sealers, so try a product called Infuze. Here's the technical data sheet. Like I said, it's a silane product that I think will work for this. It's kind of messy and smelly and dries quickly (so work as quickly); it will definitely seal vinyl seams. I think it chemically bonds to vinyl to create vinyl silane. It dries clear and doesn't discolor.