Michelle, I've never tried printing on foam core. Just off the top of my head, I'd say the main problem with it would probably be that the emulsion would tend to soak into the porous material, so yeah, I'd say you should probably size it. There's only one way to know for sure, and that's to try it without size to start, then you'll know if it will print without sizing. But I'd say the probability is pretty high that it wouldn't. That's a challenging material to start with, but it might work really great, and I like the spirit that just dives in; that's the kind of spirit that tends to succeed with gum printing. The people who want to be able to predict every step of the thing before they even start, often get so bogged down in testing this or that aspect that they never get around to making gum prints.
Gum is pretty flexible so the flexibility of the foamcore shouldn't be a problem. Foamcore isn't any more flexible than paper, is it? You can roll up a gum print on paper without hurting it at all, though it's not something I'd do just routinely. Good luck, let us know how it goes!
Katharine