Reindeer. For the last several centuries before modernity totally overtook them, the Saami had established an economic niche as reindeer herders, and before they domesticated reindeer they hunted wild ones. Also, Arctic peoples often traded with other peoples for bread, which could be stored frozen for weeks and thawed when needed.
Lapland is definitely cold and stays cold longer than southern parts, but the Saami weren't living in all that much colder conditions than your average Finnish peasant from Karelia or Ostrobothnia.
Sámi people still exists, sure, but their culture has changed a lot, naturally. They live in modern houses, use snow mobiles and helicopters for herding the reindeers (many work in "regular" jobs), buy most food from shops just as anyone else. Of course they are well aware of their roots and proud of their history.
Just an anecdote: my grandmother was born as Sámi, so I could officially have the Sámi status as well (but my kids not anymore), although I don't speak the language nor have much connections with the culture.
They are traditionally reindeer herders. There is a page on Wikipedia about Sámi cuisine you can check out. If you are interested, you could also check out the films Kukuška (2002) and Ofelaš (1987).
I loved this thriller novel: Forty Days Without Shadow, by Olivier Truc.
It's about two police detectives in Norway's northern Finnmark county. The book offers a fascinating introduction to the way of life of the Sami people and their land, Sápmi.
Apart from elk, fish, berries and a handful of hardy veggies, what did they eat?