You’ve jogged my memory and made me realize it’s even more important for OP to talk to HR than I stated previously.

I’m not sure a business license was the problem – I’m thinking it was the payroll taxes my husband’s employer was going to have to pay to Washington that they didn’t in Utah. We obviously know that employee taxes change from state to state, but don't often remember employer taxes also change from state to state.

How do taxes work for remote workers?

My husband’s employer may have accepted a pay decrease to cover their increased employer taxes. That likely would have been awash in terms of paying taxes in Utah vs. having a pay decrease but no taxes in Washington. That wouldn’t have worked for us because we were already stretching ourselves so thin, financially, to get into a pricier real estate market. We needed that gross salary as high as possible to qualify. But maybe a pay decrease is something OP is okay with. Maybe their employer is willing to pay the increased cost and keep their salary the same. That’s something OP and their HR would work out, not us.

We moved from the exact state u/1150A is living in, to the exact state they’re considering. I think my experience is very relevant here.

Since there were already a couple of other employees telecommuting, my husband’s boss didn’t realize there would be any problems with us moving and my husband telecommuting as well. He wasn’t versed in the taxes and laws, but HR was. Those problems didn’t come out until our home was sold and we were literally on our way with our children and all possessions. We hadn’t even found a home yet.

I'm trying to save OP the stress we went through.

TLDR: talk to your company’s HR u/1150A

ETA: In case it isn't clear, u/entropic_apotheosis, I've not addressed the tax question OP has asked. I'm simply stating they need to talk to HR about the logistics and make sure their home buying decision works with their employer.