You are allowed to drive your own vehicle in the monument but must stay on the dirt road. You do not need a 4x4 on it. Passenger cars drive it all the time. I was there last summer and enjoyed it. Amazing views. There is a fee to get in but I don't remember what it was. You can purchase a tour and ride in a vehicle driven by a Navajo guide.
This was taken from the Wiki on Monument Valley....
Visitors can pay an access fee and drive through the park on a 17-mile dirt road (a 2-3 hour trip). Tours are also available, and the fee varies between about $40 and $100 per person depending on the services provided and route. There are parts of Monument Valley which are only accessible by guided tour, such as Mystery Valley and Hunt's Mesa.
Over the mountains and through the woods, across the stream and up the slickrock, down the canyon and over the rocks to nobody's house I go.