There is an area in front of the Port Soent complex that is used for buggying although I have not tried it myself. Have not tried Southsea common either
West Wittering is off limits unless you are a member. This is now being enforced as members have to wear membership vests so that they can be easilly identified, also registartion numbers must be shown, helmets worn and valid insurance in place. Also, buggying is totally banned there from early April until late September.