Why can’t I FaceTime a new contact but they can FaceTime me. I am able to FaceTime 2 other contacts with FaceTime

So, when you open FaceTime, it will not open or access the camera for this particular contact? Have you tried just opening FaceTime and then selecting the contact? See this support document for assistance. If you can't make a FaceTime call on your iPhone, iPad, or iPod touch - Apple Support

If you are trying to make the call from this person's contact page, have you tried deleting the contact and starting over again? Have you had them check to ensure they do not have you blocked?

What happens when you try to FaceTime them? Do you see an error?