It depends @jschatz on how long you are working in Montana and whether you maintain a home in Oregon.
If you keep a home in Oregon you are probably still an OR resident and will have to file an Oregon income tax return. You'll also have to file a Montana return because you earned money in Montana. Oregon will give you a credit for tax paid to Montana.
If you went to Montana "for the foreseeable future" then you may be part-year OR resident and part-year MT resident in which case you would report income earned in OR on a part-year OR return and income earned in MT on a part-year MT return.
Your company should withhold MT tax if you work for them in MT.
If you become an independent contractor then you are responsible for your own federal and state taxes and should make quarterly estimated tax payments.
The IRS levies a penalty for underpayment of estimated tax if you didn’t pay enough tax throughout the year, either through withholding or by making estimated tax payments.
Generally, most taxpayers will avoid this penalty if they owe less than $1,000 in tax after subtracting their withholdings and credits, or if they paid at least 90% of the tax for the current year, or 100% of the tax shown on the return for the prior year, whichever is smaller.
Refer to Penalty for Underpayment of Estimated Tax
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