I can answer from a linguistic viewpoint. Sami speak an Uralic language, like Finns and Hungarians and some peoples further east. It's quite certain that the population now speaking Sami likely underwent a not-so-ancient language change to the Sami languages - less than 2000 years ago. But we know very little about more distant history. For that, we should consult archeology, maybe genetics. This is from an article which is a must read:

A broad body of evidence points to the conclusion that the Middle Iron Age (ca. 300–800 AD) in Lapland has been a period of radical ethnic, social, and linguistic change: in this period the Proto-Saami language spread to the area from the south and Saami ethnicities formed. Prior to this, Lapland was inhabited by people of unknown ethnicity that spoke non-Uralic languages, many relics of which survive in Saami vocabulary and place-names.

https://www.academia.edu/4811760/An_Essay_on_Saami_Ethnolinguistic_Prehistory

It's interesting that Uralic languages and Eskimo-Aleut languages are possibly connected:

Last, one can mention the generally overlooked issue that Proto-Uralic also shows some interesting lexical similarities with Eskimo-Aleut. The idea of an 'Eskimo-Uralic' genetic relationship has only been developed by some individual scholars, most notably Bergsland (1959) and Seefloth (2000), and mostly ignored by the mainstream of Uralic linguistics. It is interesting, however, that in an unbiased inspection the potential lexical correspondences between Proto-Uralic and Proto-Eskimo-Aleut do not seem to be any less promising than those between Proto-Uralic and Proto-Indo-European or Proto-Yukaghir. One could reasonably present something in the range of 20-30 comparanda, some of which even show possible vestiges of regular but phonetically opaque correspondences.

https://www.academia.edu/40193033