Can a WebWorker access the localStorage?
If not why not? Is it problematic from a security stand point?
No, localStorage and sessionStorage are both undefined in a webworker process.
You would have to call
postMessage() back to the Worker's originating code, and have that code store the data in localStorage.
Interestingly, a webworker can use an AJAX call to send/retrieve info to/from a server, so that may open possibilities, depending on what you're trying to do.