I've seen this done a lot with twitter at a lot of real world conferences I've been at. Simply come up with a twitter hashtag people should use to direct questions towards you. Other than that, perhaps Google Hangouts???
A great way to do webinars is with Chatwing and Google Hangouts. Basically you use google hangout for the video side and then chatwing for the chat side. There are several advantages to using chatwing, 1 of the big ones is that chatwing has 10 different login methods you can choose to enable so if you are doing a business type webinar you may want to only allow users to login with linkedin. or if its about twitter then you can require only twitter login to chat. The nice thing is you can choose as many or as few as you want so its ususually a good idea to allow google login especially if you are using google hangouts.
There are a ton of other features you can customize within chatwing including the ability to upload your logo for the webinar plus add moderators access and control the chat from the chatwing mobile app.