The entrance to the cave is expressly signed that the hikers ARE NOT to bring food stuffs OR hiking staffs OR leave any debris that may be left behind and damage the ecology of the cave OR bring pets OR have open flames. You are also not to defecate or urinate in the cave so facilities are provided above ground. You are NOT to touch the walls lest you damage the ecology of the cave slime.

So you are not allowed to have them at all, or just using them in cave?

As per the website:
http://www.fs.fed.us/gpnf/recreation/ape-cave/respect-the-cave.shtml#protect
My interpretation: if you bring food, don't eat it in the cave.
The basic rule is there to prevent snacking, gum/junk food, wrappers, debris, etc. Nothing says you can't take food with you and have a picnic outside at the upper exit of the lava tube!

It isn't a long hike. I see little reason to bring a bunch of food with you. Water up and then "de-water" before going into the tube because if you really have to go, then you need to be on outside of the lava tube.