Site Admin here.
I can't really answer your question with 100% certainty as I have not hosted a chatroom service before.
Here's Hostgator's article about the difference:
Why should I use VPS instead of a shared or reseller package?Since the site and forum's format does not require intensive processing and is definitely not resource-heavy, it works very well with Shared Hosting.
I imagine that depending on the processing load of whatever chatroom application you will host, Shared may or may not be appropriate due to resource sharing with other clients on your Server. VPS should be more flexible for that purpose, although I don't think you'll need a Dedicated Server unless you'll be looking into thousands (if not tens to hundreds) of participants all going at it at the same time.
I suggest you contact Hostgator's pre-sales support so they can answer your question much accurately than I can.