When will Stafford County start excepting refunds?

The IRS does not begin to accept e-files until January 29.  A state return cannot be accepted until the IRS has accepted your federal return.   

You do not say where "Stafford County" is---we do not know what state that is in.  You might be on your own to get the forms and mail them to the county.

TurboTax only prepares a few city or local tax returns as an extra step in your state return.  For most local tax returns you are on your own.  You may have to ask around amongst your neighbors, etc. to find out if you have a local tax and how to pay it. “Google “ local tax for your area.  Call your city government, etc.  Your only record of paying your local tax will be your own bank/credit card records

https://ttlc.intuit.com/questions/1899325-which-city-tax-returns-can-i-do-in-the-turbotax-state-program