The specific cooler seems to be a model 5206, which was introduced in 1959 and 1960 and shortly thereafter discontinued, when Coleman introduced the full "Snowlite" line and began making them with a white base later in 1960.
Note: A similar cooler is seen is season 8, but they are slightly different models. We can tell this model was made between 1962-1964, because it has the diamond imprinted on top, but it has the "cam-latch" which was introduced in 1962. By 1963 all coolers were manufactured with it.
Dean grabs some beers while they discuss their current hunt.
The cooler is seen inside for the first time. Dean digs out a bottle while talking to Raphael.
The cooler is seen inside the Impala. Post helping the townspeople to take care of some demons, the boys grab a celebratory beer out of the backseat.
Dean drinks a beer after (soulless) Sam refuses one.
Dean has his feet propped up on the cooler before getting himself and Sam some beers. Later, Sheriff Mills arrives to help Sam find his brother, the cooler staying supportively in the background, steady in this time of crisis. (Method Acting).
The cooler is brought inside, and Dean moves it closer to the sofa so he can enjoy a drink.
A green cooler similar to the boys' one appears in Benny's camp.
A later model of the Coleman green cooler is seen in Rufus' cabin. These started to be made in 1961.
A similar model green cooler is seen in the vampire's house.
Sam and Dean relax with some beers out by a lake, with the green cooler between them.
Claire hangs out with her new friends - the ax murderers. @TheGreenCooler says he was asleep during this scene.
The green cooler holds beer, smoothies and a decapitated head.
Dean has a beer and sits on the cooler while letting Sam finish digging up a grave.
Sitting quietly over by the sink.
The cooler appears in Heaven outside of Harvelle's where Bobby and Dean share a couple of beers from the cooler.
The cooler is in the Impala's backseat as Dean drives away.