I'd say you're good to use the wood if you seal it (I'd seal it even with plexi over it). I haven't seen you say what type of wood it is, but chances are if its regular plywood then it is pine. Instead of plexiglass over it, you could also do vinyl tiles or something... I suggest that because its much cheaper and easier to cut to size than plexi
As far as sealing it goes... Mod Podge is junk and the seal is "meh" at best (personal experience). Its ridiculously over-priced elmer's glue, and if moisture sits on it, it will soak into it and stain. If you're wanting to go cheap then seal with watered down wood glue (watered down for sake of application). Couple coats of this is a much better seal than an equal amount of Mod Podge. If you're looking to go a little higher quality then I'd suggest a water-based Polycrylic or something similar. I know people have also used marine varnish (wood sealant for boats) and such too...
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