I think the basic idea here is a great one. However, some of the finer details have me a bit concerned. I am not familiar with this series of webinars, so if part of this me not understanding please correct me!
It would be great if the webinar started with a brief introduction to the site. Stack Exchange sites are not traditional forums, and sometimes this can trip up new people. When it is a user here and there, it is easier to give them attention and help them learn how to use the site. When it is thousands of people all at once, it can easily be chaos. Even a few minutes explaining the premise of the site (not questions just for discussion, but about objective problems you're facing etc) would be helpful.
What sort of questions are people usually asking in the chat window during these webinars? Are they questions about the presentation itself? Are they discussion questions? While promoting the site is great, don't lose site of the purpose and intent of the site.
I wonder if this is essentially trying to treat the site as a chat room. Remember also that there is the chat room (and more chat rooms can be created if desired). For general discussion and chit-chat about the webinar, this could be appropriate. However, one must have 20 rep (on any Stack Exchange site) in order to chat.
I'm also a bit curious about the tag. If the question is truly a good fit for the site, then a definition-of-done tag doesn't seem like it would fit the question. definition-of-done would seem to serve the purpose of knowing what questions are from the webinar, but seeing as this site doesn't get Stack Overflow's 3600 questions a day, it is probably ok to assume any question asked during that timeframe is from the webinar. Using the active or new sort would show these questions.
Note that if you do use chat, a moderator should create an event in the chat system. This will publicize the event via a system message on the site, and also tweet the event to the site's twitter account.
Don't let me scare off this idea - I do think there is value here. I'm just curious about a few of the details.