Do I have to withhold Virginia state income taxes on my nanny if she request she does not want it withheld by me?

See attached on VA website

https://www.tax.virginia.gov/withholding-tax

Probably.  If you meet the VA definition of an employer and are required to withhold taxes, then yes, you must still withhold her VA taxes even if she does not want you to.

You are considered an employer by the state of Virginia if the following apply:

• You are a resident of VA
• You are doing business in VA 
• You are paying wages or salaries for services performed in VA