There's always a way. Call in plumbers to see what it'll take to do it right and get estimates on the cost.

"Is there any other way for the water to drain or for it to get water when the washing machine is not near a water supply?"

Seriously?

I’d like to move the washer and dryer up to the main floor of the house

As in up from the basement?

If there is unfinished basement below, it should be relatively easy to accomplish.

I cannabilized a bathroom in a condo to make room for a stackable washer and dryer years ago. The plumbing was easy; it was getting the 220 line from the breaker box in the basement to the third floor that was tricky.

No, you need plumbing and/or additonal gas or electical outlets to run the washer and dryer. You also need a dryer vent.

There is no way to use plumbing that does not exist.

Given the cost for small plumbing jobs, I can only imagine what doing this must cost. You will likley not only need a plumber, but also the services of a carpenter and an electrician.

That looks like a living area in the photo. Is that where you want to put the laundry?

I remember way back when we had some washers that just drained under the house. Really. That would have been a house built on piers and somewhat rural. And people did not use to wash as much laundry as we do now, or use the chemicals that many now use.

Not a suggestion.

No, you cant get water without plumbing.

If you could situate in or near a bathroom that allready has some of those features, or a space in the kitchen near the facilities that would be your best bet.