Need to lay a new laundry room floor

I would suggest hiring a neighbor/teenager who is strong enough to handle. Move from side to side until floor is finished and move back. $20 might be enough for the help. If either machine ever needs to be serviced or replaced lifting them from old to higher level if new will be a hassle. Plus cleaning old vinyl under and behind will be a nightmare.

We used vinyl plank flooring. It looks like wood floors but is water resistant. It‘s a floating floor so nothing is glued on the existing flooring and can but put down without removing the old floor. They have 2 types, a click-it type flooring (almost like a tongue-in-groove) or flooring where each plank has a sticky edge to press against the another plank.

I would chip out the old vinyl. If you can prop the washer and dryer up a little to get part way under it that would help. You want the floor to be level, and probably don't want to see your pretty new tile, with the ugly old sticking out around the washer and dryer.

Borrow some help or pay a neighbor kid a few bucks to help move the machines out and then back in. Any spills or over flows will be a breeding ground for mold under the machines seeping into and between new and old flooring.

As others have said, it's not that hard to move the machines. Most come with hoses that are long enough to move them out of the way without disconnecting. If your flooring is in pieces, like a tile of some kind, you can probably do it without disconnecting them. In the long run, it will be worth the effort.