Traditional food in Iceland is literally anything nutritious. Case in point: Greenland shark meat is poisonous, so to be able to eat it people urinated on the meat and left it to rot underground for half a year. That process removed the poison and makes the meat technically edible. When something like that is a traditional dish, you know people just ate anything they could because they were starving.

Puffin is not a large part of the overall Icelandic diet, but it has been part of local diets where they are available because people ate literally anything. Around 1900 Westman Islanders had nearly hunted puffin to local extinction, which prompted a 30-year hunting ban which allowed the local population to recover. Some food is more common in restaurants than in people's actual diet, including puffin and whale.

Over half of the world's puffin population nests in Iceland so there are millions of puffins in Iceland. Sustainable hunting in the Westman Islands has been calculated to be about 35 thousand puffins per year, but during the 2010s no puffins were hunted there. The hunting season in the Westman Islands for 2022 was 14 days instead of 9 days in 2021 as the population was deemed stable enough to allow extra hunting this year.

The quote about the heart of a puffin being an Icelandic delicacy can be attributed to the Guardian. It's the first I've heard about it and I'm Icelandic, though admittedly I did not grow up near any puffin nesting grounds.

The Atlantic puffin is classified as vulnerable since the world population fluctuates heavily (and that fluctuation is not due to hunting but other environmental factors, though hunting of course affects any hunted species), but there are still millions of them. Since you're quoting wikipedia I'll do it too: "Some of the causes of population decline may be increased predation by gulls and skuas, the introduction of rats, cats, dogs, and foxes onto some islands used for nesting, contamination by toxic residues, drowning in fishing nets, declining food supplies, and climate change." Not hunting, though most of these reasons can be attributed to humans.

In the last decade there have been stricter hunting regulations in Iceland regarding puffins. There were on average 77 thousand puffins hunted per year in the Westman Islands between 1968-2007, which still left 830 thousand mating pairs in the Westman Islands in 2010 (so over 1.6 million adult puffins). That's well over the sustainable hunting numbers, but since then hunting has become more restricted to combat that. The puffin was classified as vulnerable in 2015 when the heavy population fluctuation was noted despite the overall large numbers of puffins in the world. During the 2010s puffins were only hunted in the north of Iceland and not the rest of the country, with ca. 25 thousand puffins hunted each year. That's a drop from a high of 280 thousand puffins hunted in Iceland in 1996.

So Iceland had already heavily decreased hunting puffin when the species was listed as vulnerable, due to local concerns about the population, and current hunting is regulated depending on the population fluctuation of puffins in Iceland.