Armadillos can inflate themselves by swallowing air and bloating up the intestines. I have to think this takes some time. http://www.rainforest-alliance.org/species/armadillo
They famously do it to be more buoyant when swimming. Armadillos can also wedge themselves immovably in tunnels and I would not be surprised if the inflation helps with that too.
There are lizards which inflate themselves for various reasons. Chuckwallas wedge into cracks and inflate themselves so they cannot be removed.
Toads (also frogs, I think) inflate themselves to make themselves harder to swallow. Also to avoid sex if not in the mood.
I am not sure what mechanism these use - probably also swallowing air.
Rather than swallowing gulps of air, if an animal were rigged so it could close its mouth and nostrils, it could use its respiratory muscles to exhale and force air down into its stomach and intestines. That would allow rapid inflation. The animal would still not look like a puffer fish; it would look like me.
Very interestingly, especially in light of above hedgehog discussion, when I googled "subcutaneous emphysema" I got a bunch of hedgehog pictures. ??
It turns out there is something called Hedgehog Balloon syndrome in which damage to the lung allows air to fill spaces beneath the skin, puffing it out just like a puffer fish. An out of control, sick, wounded pufferfish.
Subcutaneous emphysema can happen to humans too and it is always the result of a wound: abnormal connection between subcutaneous skin and lungs. If these hedgehogs had a such a communication like a sphincter that they could relax and open, they could inflate themselves with a valsalva maneuver by increasing pressure in the chest but not allowing air to escape the nose/mouth. Inhaling against a closed glottis would create negative pressure in the chest and suck all that air back in.