Call the resort and ask. I can not think of a reason you couldn't. We bring our own bikes and fishing poles. You will have to find a place to put in on the Island. I'm sure they have public access areas around. All you need is a map. When are you going? We'll be there the last week of July.
I would call and ask, because you don't know if there is a fee that has to be paid. Kinda like a fishing permit. if there is one, you would not want to get caught with out the permit.
I was at a river once that had an honor system pay box and the Rangers do check. it's a $500.00 to not pay the $1.00 river pass.
Yes, you can launch your own kayak at the far end of the resort near the harbor inlet. There is a picnic area there with a small "beach" suitable to launch kayaks or canoes. In the past, the resort kept kayaks there for resort activities and launched from the same spot. Those programs were right oin the marsh behind the resort.
There is a parking area there and many just leave their kayak rooftopped and launch whenever they want.
I'm not sure what is needed for fishing licenses - none are required for the fishing excursions on the island.
I have friends that always arrive at the resort there with 2 kayaks strapped to the roof of their Suburbun. What's better than owning your own kayak? Having friends that do!