You'd need a "C" harp to play "cross-harp" or blues-style in the key of G. I'm pretty sure that's the approach used for the song in question, even though it's nominally "country" and not by the strictest definition "blues."

I'm as shocked as you are if a basic non-chromatic mouth harp costs 20 bucks these days, but I wouldn't be surprised. That would mean that a full set, enough to play in any of the most common keys, would cost well over $100 -- bummer!