If you need mental health services through the VA, can you use a private doctor or does it have to be directly through the VA?

The VA does provide mental health care services to veterans for many (but not all) mental health problems. Depending on your needs, I suspect the VA will likely provide services to you directly for free, but they won't pay for outside providers unless you are enrolled for VA care, the mental health need is service-connected, they don't have capacity to see you, and you will use one of their preferred providers. It's not as open-ended as "Go wherever you want for whatever services you need, and send us the bill."

Have you reviewed this resource page?

https://www.va.gov/health-care/health-needs-conditions/mental-health/