How do I make a greeting card into a plaque?

Mod Podj (can't remember how to spell) would be your best bet.

I'd look for a blank wood plaque at a hobby store, craft store, or big box store like Michaels. Then, as Heje said, use Mod Podge to coat the card and adhere it to the plaque.

I think it would be cute to attach a picture frame to a blank wood sign and slip the card inside the frame.

I would buy a mat to frame it and put it in a picture frame instead. Get an 8 x 10 frame and a mat inside. No fuss.

I think a nice frame or even a shadow box would work well for this.