I've got to agree with Buck and Lifey, that the free discount cards are very worth it for all visitors. For stores that have them, you can't get sale prices without it, and that means you could easily pay nearly twice as much for your food!
I can only think of one chain that has ever mailed me a card, and even there I got a paper temporary card. The vast majority just have a big stack of plastic cards that they just activate on the spot. Some stores will do a courtesy swipe if you are just passing through,
but it is hit or miss (the larger the store, the less likely they will do it in my experience) and so I just quickly stop by the service counter and grab one.
It is true most grocery stores in the US are regional chains, but many of them have the same owner. For example, I have a "City Market" card, but that is really just one of Kroger's brands so I can also use it at Krogers, Frys, Ralphs, etc depending upon what part of the country I am in. You can do like lifey and ask if your card works, or a lot of times you can look at the stores generic brands and get an idea if its is the same parent company.
One final thought on why Roadtrippers should look at the Grocery cards is that many of them also can get you discounts on gas. Some grocery stores will have their own gas stations, and the card will save you there. Others will partner with local gas stations, and you can get discounts on gas based on how much you spend, or if you buy some specific items.