Either version would be OK, Atticaunt. You could use Puffins adjectivally, i.e. the class of the type, "Puffins", or the possessive plural, Puffins', because it's the class of/used by/reserved for the Puffins. I prefer the latter, but others here would probably suggest the former.
I'm going to muddy the waters. I think we're barking up the wrong puffin here: both Puffins Class and Puffins' Class sit uneasily with me. But it's not an apostrophe issue, it's a singular/plural issue.
I'd prefer Puffin Class. What are the other classes called?
My local primary school has Oak Class and Beech Class; cubs and scouts have Badger Troupe and Red Six.
Oaks Class, Beeches Class, Badgers Troupe, Reds Six all sound very odd, and bunging in an apostrophe isn't going to help!
I'd say it's definitely not possessive: Wolverhampton Girls' School would be a school for girls and Puffins' Class would be a class for puffins rather than children. So it must be adjectival, and although we have sports car and accounts department, they are unusual. Department stores have a shoe department, not a shoes department.
I think Paul has clarified things a lot - it's like the difference between "Football Club" and "Footballers' Club", it just depends what is in your mind when you say/think it. From that analogy, I think, like Paul, that "Puffin Class" works much better.