You do NOT need to be a member of MBBA to use the beach. You do need a permit issued by the Oversand Permit Office at the Cape Cod National Seashore, which is located at Race Point Beach in Provincetown. You can buy a seasonal permit for about $150, or a weekly one for I think $50. The rangers will inspect your vehicle for all the proper equipment, including the right size tires. The biggest hurdle with the Element is that a full size spare tire is required. The sand is soft and deep in many places, and many of the beach access points are fairly steep. I've seen CRV's on the beach, and have heard of a somebody with an Element, but never seen one. I also have a Toyota 4 runner which handles the beach really well. In an Element you'd be dragging the center pretty much all the time.
The info you need is here: