I don't think this is unworkable at all, although it might push some edges a bit, depending on people's past experiences. I have three concrete bits of advice and a couple suggestions:
Stay mostly in one place - Make sure there's a solid default location, a stable holding with a PC hardholder. This allows for lots of comings and goings as players inevitably can't make it due to real life, or as characters have different interests. A mobile group of traveling PCs is *far* more difficult to run for with a group that big, because you've got to pretend some PC's right there when the player is actually working late.
Move the spot-light around - If you've got more than 4 PCs, don't let any of them roll twice in a row. Move at the start of the action, in the middle, or at the end, and vary it.
Make relationship maps and take notes like crazy - However works best for you, and for session notes, best if it's not you; figure out which of your players is most naturally a note-taker, and delegate that to them. With 7 PCs, you're going to have some complex webs and connections, and tracking them will give you stuff to fall back on and interesting angles to exploit. This will also make your prep *much* easier.
Consider co-MCs - If making fronts that connect with so many PCs feels daunting, ask someone to help co-MC, especially for the prep work of creating fronts. If combat makes you anxious, ask someone else who's really solid on it to help MC those bits. Also, be really up-front about asking everyone to barf forth the apocalyptic details.
Don't sweat it - A group this large, someone's bound to be absent, or someone will have to drop out. When that happens, just roll with it. We were all sad to see Jodi's real life commitments increase and her really cool Hocus PC fade away, but that's how it goes. Move on with who's there.