Here's info on on IBSP beach driving:
Island Beach State Park, New Jersey, driving on the beach
Looks like you can get a 3 day permit for $50, valid during the summer season (it's likely they consider "summer season" to be from Memorial Day to Labor Day in the US, so last weekend to early September, but it is possible they mean the official start of summer in late June) or a yearly pass for $195, which obviously you would not want.
Here's another link including LBI towns, but it looks like most of them deny driving access from mid-May to mid-September.
New Jersey Beach Buggy Association This link also includes more specific info on Island Beach State Park.