How can a qualify for a tax credit to live in Oregon but work remotely in Utah to avoid getting double taxed?

Just about all states have a reciprical tax agreement of some type. Whats specific tax credit are you expecting?

One important thing to keep in mind when you have to file more than one state tax return, is to do your resident state return *LAST*. That's the only way the program can *CORRECTLY* take into account any reciprocal tax agreements that may exist between the states being filed.

I't also important that you not file *ANY* tax return (including the federal return) until you have completed *ALL* returns and are satisfied with the results. When you do your resident state return last, it *WILL* change things on the federal return, and depending on the specific states being file, it *may* change things on any and all non-resident state returns also.

Very helpful thank you!

If all your remote work is actually done in Oregon, and you never physically work in Utah, the income from that work is not taxable by Utah.  (But it is fully taxable by Oregon.) 

Some states tax non-resident remote workers, but Utah is not one of them.