Well, to start with, $350 is 1,400 quarters. Laid end to end, they'd stretch 111 feet -- but that's not the most practical way of stealing coins.
How much do 1,400 quarters weigh? Well, one quarter weighs 0.2 ounces, so 1,400 clock in at 280 ounces -- which is 17.5 pounds. That'll last a long time at the videogame parlor, but you don't need a flatbed to haul it away. It should fit nicely in a large backpack. The Safeway right next to the laundromat in question doesn't have a Coinstar -- and staff there didn't remember anyone with a sack or pack full of money dropping in recently. But plenty of area stores do. You wouldn't even look out of place toting a massive bag of coins; everyone using a Coinstar does that.
Sad to say, but barring a near-miracle, it looks like this laundromat thief got away clean.