Looking at this you might be thinking - what do we do with the Web?
I envision that it would become part of an interagency portal, much like Amazon.com.
The portal is an IT function and thus would be administered by the Federal Chief Information Officer. Each agency would be deputized to contribute to this portal.
Generally each individual agency public engagement function would answer to or be part of a larger interagency task force, group or committee - whether it's FOIA, social media, media relations, writing and so on. This is the concept of shared services as applied to citizen communications as well as the concept of a customer-centric organization organized from the outside-in.
The Amazon storefront is an example of a portal that pools individual vendors who retain control over their presence into a highly customer-centric virtual storefront. (In the real world it is akin to a mall storefront.) The vendors cooperate with Amazon's standards to be there, but are individual presences as well.
The first thing you see on the Amazon portal is a search bar. You can search any "department" you want. In the government's case you could search any agency or subagency (you could mouse-over the agency and have the sub-agency be a set of choices within it - similar to USAJobs).
Of course every Agency is concerned that it have an opportunity to tell its story its way. Which is why each would get an Amazon-style "storefront."
Finally of course there is the "help" function, which would be an interagency function. That way if I ask a question I don't have to think about which Agency would answer it. It's the job of the individual Agency help desk to participate in the system and ensure a response.