Just as a head's up, on January 12th, Flash Player is no longer going to load content from the open web.
I see questions from JPay users on a regular basis, and I'm always somewhat surprised (and somewhat not) that they're still requiring Flash Player. We announced in 2017 that it was going away (along with browser plug-ins in general) at the end of 2020. Background: https://theblog.adobe.com/adobe-flash-update/
Given the potential impact to the families that depend on it, I've been wondering about what their transition plan is. As a paying customer, you might want to reach out and ask. At the very least, if you know that there's a technical train-wreck coming on their end, you can at least prepare your son for it.
For the more immediate issue, if you haven't rebooted the laptop, that's a great place to start. Make sure that you have the latest Chromebook updates installed. We definitely haven't changed anything in Flash Player recently with regard to webcams.
You can click the lock next to the address of the website to see if Camera/Mic and Flash permissions are allowed, and if you right-click on the Flash content and choose Global Settings, you should get a webpage that has the camera/mic settings for Flash Player in the context of Google Chrome. Unless someone accidentally clicked deny on a permission prompt, it's unlikely that this will help.
If you had another tab going in the background that maybe was keeping the camera busy, that might be what's going on. Rebooting should solve that, too. If you have an adblocker or something, you might try disabling it.
If none of that helps, here's a random webcam test: https://www.onlinemictest.com/webcam-test-in-adobe-flash/
It would be interesting to know if it's all webcam activity in Flash, or maybe something specific with JPay...
Hope that helps!