Can I sue someone for bitting me, as well as my employer since this happened while at work?

Under Virginia law, you would be able to sue the person who bit you but you would not be able to sue your employer, assuming that you were at work and performing work duties at the time of the incident. Your rights against your employer would be under the workers' compensation statute,  which provides an easier way to prove liability but allows more limited damages. 

In cases such as this, an experienced lawyer will also look for any other possible defendants, so the options here may not be limited to your employer or the biter.

You would have a valid claim against the person who bit you, although it is unclear what defenses might be raised to the claim.  What is equally important is how much injury (both economic and non-economic) was sustained.  For a lawyer to take a case such as this on a contingent fee basis, there would need to be sufficient medical bills, wage loss and possibly permanent injury.  In other words, although you have a claim, you and a lawyer would need to evaluate probable liability and probable damages to decide if it is worthwhile to pursue.

It really depends on the facts of the situation in which you were injured.  You definitely have a workers compensation claim which should cover your medical bills and any lost wages you may have had as a result of this injury.  Yes, you can sue the person who bit you.  However, you have to keep in mind that bringing a lawsuit against someone is expensive and time consuming and even if you are able to obtain a judgement in a court against the person who bit you, it doesn't always mean you can collect from them.  Whether or not you have a claim against your employer really depends upon the facts of the situation.  If you can show that your employer knew that security was needed at your place of employment and failed to provide that security, you may have a claim against your employer.  You need to be able to show that your employer owed you a duty, that they breached that duty and that their negligence caused your injury.