Can I be paid as my mother's sole caregiver in Virginia?

You can have a contract with your mother to be a paid caregiver.
No training is necessary to be paid as a caregiver.
But a contract should include ..and I am sure others will include some that I miss.
Taxes should be taken out
Hours should be consistent with actual working hours. (no one works 24/7/365) time off has to be scheduled.
I would check home owners insurance to be sure you would be covered if you were to be injured.
If she is living with you it might be easier. You can charge her "rent" a portion of all household bills. This would include mortgage, insurance, water,sewer, gas, electric, cable and anything else. This might actually be easier than a contract and having to pay taxes on your income.
But there should be someone else that can work a few days a week to give you a break. A caregiver 4 to 6 hours 3 days a week makes things so much easier.

I have a friend who went through CNA training to become his mother's paid CG. I do know he had to do this to be paid through her Medicaid. A cautionary tale--he does not make much.

Worth looking into.

Also, mom must have some funds. They should be spent in her care--and you should be paid, somehow.

Keep records, so siblings and others can bust your chops about it.