I’m cracking up picturing all of the dopers in this thread (doctors, scientists, and such) sitting in front of their computer doing their best impression of a chipmunk. 
Oh, you can also sorta do it if you have someone push his/her lips against yours and blow. Takes a bit more skill, but certainly much more entertaining for you and your SO.
Isn’t it the Glutieus Maximus muscles that are used?
That is exactly what I was doing; and somebody walked right by my cube in the middle of my “experiment” and gave me a funny look.
Now I feel like a doink.
Sorry, a late reply, but I don’t think anyone has answered the OP correctly.
The muscle used to blow out your cheeks, such as when playing a trumpet or blowing through a straw is the buccinator.
Here (a few paragraphs down) is a brief description. Here is another, unique, blurb.
That was my first thought, too, KarlGauss. It’s certainly the classic Gross Anatomy answer (Most of ut were shown a slide of Dizzy Gillespie or some similar classic jazz trumpeter at some point), but strictly speaking, the buccinator does not puff your cheeks at all; it merely compresses and/or modulates the pressure of the air when your cheeks are puffed.
After all, unlike the complex interweaving of fibril in the tongue, the buccinator can only contract uniformly. Its origin and insertion don’t offer it any leverage to generate forces to make it “bulge out”. It is a sheet muscle, so it can’t even “make a solid bulge” in the limited manner of bulkier muscles like the masseter or biceps.
The buccinator doesn’t blow out your cheeks; quite the contrary, it compresses them. When playing brass or blowing (note: not sucking) through a straw, it acts to compress air by reducing the amount the cheeks puff. This is why so many people suggested trying to exert the force of many of the proposed candidates without air trapped -by whatever means, or to whatever extent- in the oral cavity.
Perhaps you have exceptionally talented buccinators. The rest of us don’t, yet “puffing one’s cheeks” is a fairly universal skill.