You can find all kinds of sizes of Scotch envelopes to send oversized cards in. They are in the "home and office" section of Target, Walmart, and even grocery stores. For your size card, they cost about $2. They are stiff and bubble-lined, and I just write "Do Not Bend" on the front.
I make oversized cards all the time, and I use these envelopes to mail them in. Never had any kind of problem.