Peter is the cuddler of the marauders and he gets the others hooked on it.
James was an only child and while is family were close and loving, they just didn't really cuddle. Remus was always afraid of what he was to even think about cuddling with his mom, what if he hurt her? And Sirius is just obvious. The Blacks weren't the cuddling type.
It starts with James. The Pettigrew' move in just down the road, across the small woods and meadow. The two instantly hit it off and Peter is a very tactile person. He's holding James' hand any chance he gets. Excited little hugs when they see each other again.
They have their first sleepover and Peter just, scoots over in his bed for James. Halfway through the night, they end up cuddled together. It becomes a tradition, a norm for them.
So when they go to Hogwarts and become friends with Sirius and Remus, Peter doesn't hesitate to invite them too. It takes time, some coaxing and promises, lots of trust.
By the end of second year, they can be regularly found cuddled up together. Peter and James, out on the Quidditch bleachers, side by side. Sirius and Peter's desks pushed together, arms wrapped around each other in class. Peter and Remus cuddled up on the couch as they study or read or nap. One big cuddle pile on nights leading up or after the full moon.
Cuddle bug Peter infecting the rest of them.