Nope. No. Definitely not. You should not use your social security as the tax ID number for your single owner LLC in Virginia. Yes, I know, the IRS allows you to do that. They do not require a single owner LLC to obtain a separate Employer Identification Number (EIN) and they will allow you to just use your social security number. But it's a bad idea.
It's a bad idea because of the risk of identity theft. As you grow your business, your tax ID number for your LLC will have to be shared with many different people and companies; vendors, customers, suppliers, and the list goes on. You do not want to have your social security number floating out there everywhere. So get an EIN from the IRS for your LLC. It's free and available online. And then use that as your tax ID number and keep your social security number private.