Any State can legally tax you on income earned there; some are more aggressive than others. File a VA non-resident return, and you may get some of what was withheld back. Don't file, and you'll likely get a letter from VA asking why you didn't ...
You say where your employer is located but not where you live. I assume that is also FL.
The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state. The fact that you live in a state without an income tax does not get you out of paying tax to other states.
For more info, see http://money.cnn.com/2013/06/28/pf/taxes/business-traveler-tax-threat/
your situation is like that of pro athletes who have to file tax returns in most of the states they play games in + if they don't live in the same state as the team, then they also have to pay taxes to their resident state. their resident state does allow for credits for the taxes paid to the other states.