Employer Won’t Withhold Taxes

Hypothetical… employee works remotely in state A. Employer has been made aware but forces them leave all of their paperwork and residency assigned to state B. The employee technically should be paying state income tax to state A, but state B has no tax so none are withheld from their paycheck. 

Can the employee pay tax for the year as a lump sum to state A when they file? Will there be penalties for this? The employee notified the employer numerous times and requested a change but the employer refuses. So the employee either pays no tax or does… what? I don’t know.