My employer's payroll company did not withdraw the correct percentage of state taxes from my year's worth of paychecks. I'm now liable for $2,000 in state taxes. Can the payroll company be held fully or partially liable for this amount?
No, a payroll company isn't liable for paying tax you owe. They calculate how much should be withheld from each paycheck and direct your employer to pay it to the tax authority.
The amount withheld isn't always a perfect match to your tax liability because, because you may have other income -- like from other companies, interest income, or stock sales.
Withholding is controlled by the W-4 at the federal level, and similar form at the state level. Review your elections on the W-4 form to make sure they reflect your family tax situation.
In the highly unlikely situation that the payroll processor made a mistake, and that resulted in you owing underpayment penalties, you might have some luck in getting them to pay the penalties. But I doubt you are in that situation. And regardless, you are personally responsible for your own tax bill.