Many other StackExchange sites have a logo on the traffic page. How can we go about getting one of our own, as opposed to the default blue chatbox and M we have now?
Monero Stack Exchange is still in public beta. A site goes through several stages from proposal, definition, commitment, private & public beta, to finally the graduation (explained that on reddit a while ago).
We are currently in public beta, this is the last and most important step before graduation. Your follow-up question will be probably, what do we have to do to succeed and graduate? It basically breaks down to a strong community both in terms of active users and frequently questions asked per day, about 10 per day on average.
Custom site themes and logos will be applied after graduation.
At first glance it looks like the sites with a logo all have graduated beta. If that is a requirement then we probably do not have the option yet.