This site seems to have all the info. I saved it a while back but never had the time to read through it or head to the beaches.
http://www.libba.com/index1.htm
From what I understand most of the 4x4 beach permits are intended for people who are going to fish on the beach. Some of the permits will require you to a watch a film on nesting birds circa 1970. Almost all of the places will require that you bring and use your fishing gear and poll. Here’s what’s on the site:
Check list of equipment needed when you drive on any Suffolk County beaches.
Air Pressure Gauge - min. 5psi ( low pressure )
Air Pump (recommended not required )
Jack
Jack Board 3/4" thick x 12" x 12" (min.) hardwood or ply
Lug Wrench (recommended not required )
Shovel
Tow Rope or Chain
Flashlight (recommended not required )
Minimum of 1 red flare
Minimum of 1 fire extinguisher (USCG Approved)
First Aid Kit
Plastic Pail (recommended not required )
Spare Tire
Garbage Bags
Porta-Potty
If you’re driving on the sand I’ve read that 235 tire width is the min you want although I’ve done it on 225’s in my old Cherokee with the tires aired way down. I haven’t had a chance to go in the Liberty yet but I would like to. If you pick a weekend to head out let me know and I’ll try to make it.