Unfortunately, the only webcam currently working on island is the Sunset Beach Bar webcam, which is at the end of the runway. Not really a beach that most people go to for the BEACH though. Link
There used to be a couple on Orient but both are gone, wiped away by Irma, unfortunately.
there's also the Simpson Bay Beach Resort webcams. There are several and one is of the beach area. There's another one of Great Bay in Phillipsburg, just google Sea Palace Resort and click on the webcam.
I guess people who are not staying on property can use the Simpson Bay Resort chairs?? Don't know. I personally wouldn't feel comfortable doing that. I think the original poster has already been to Great Bay..