I wish I could help you more but I'm not a lawyer. To explain it a little more, there's no federal law (that I know of) around this topic. It's likely a state law, if any exists at all. I'm basing this train of thought on recording phone calls. Some states are illegal with only one party consent, some say you only have to get ONE party's consent (yours) to do so.
As a layperson, my guess would be no it's not illegal if you record your own camera stream. Can it be done? There's places online that claims it can but I'd guess you're breaking a TOS (terms of service) in doing so. Which may, or may not matter. IMHO, it comes down to using something for a purpose that it's not intended. Not necessarily illegal but you'd likely catch flack from the company if you ever took it to them.