Most states require applicants for drivers licenses be able to produce documents proving both their identification and also their legal Immigration status in the US. The Virginia Department of Motor Vehicles has a comprehensive list of both types of those documents, and other requirements, such as proof of VA residency and a social security card. You can download that list here. You should read this very carefully.

If your VAWA case is based on Marriage to a US citizen (as opposed to marriage to a permanent resident) you can file your I-485 adjusment of status application together with the I-360 VAWA self-petition. Applicants with pending I-485s can also simultaneously file for employment authorizations, which would give the ability to work and get a social security card. Virginia requires you to prove you have a social security number, so you'll need to be authorized to be employed to get a drivers license.

For proof of legal immigration status, VA's list is comprehensive, but most forms of proof relate to status as a permament resident or asylee. During your process, you will only be an applicant for permanent residence, so most of those forms of proof will not be possible for you. They do accept certain I-797 "Notice of Action" forms, but only certain ones. They do not list which ones. My belief is that if you have a notice of action that indicates some aspect of your case has been approved, such as approval for employment authorization, that would be enough.

So the steps you should take before applying for a VA drivers license are: file your I-360 with your I-485 and I-765, wait for I-765 approval and your receipt of your employment authorization card ("EAD" or "work permit"), apply for a social security card, and then bring everything, along with your proof of identity documents (your passport and marriage certificate look like ones from the list you should be able to produce), and then apply for your license. It should take around 2-3 months to get the EAD back after filing your applications with USCIS.

Note that this procedure is if your VAWA case is based on marriage to a US citizen. If yours is not, you'll have to file your I-360 alone and wait until the priority date becomes current before filing your I-485 and I-765. This will significantly delay when you would be able to get a drivers license.