The Sami never had the numbers, the organization or the technology to create a nation that could rival a Swedish, Norwegian or Finnish power. One of the main reasons is their nomadic, reindeer-herding lifestyle. And that is due to the fact that they live so damn north they can't really do anything else for a living, as a people that is.
The Sami used to enjoy much bigger living areas in Finland and Sweden until the 16th and the 17th centuries, even, before the southern peoples practicing slash-and-burn agriculture moved north and took over those areas. Central and even parts of the southern Finland have a lot of place names derived from the Sami language due to that. And there were very small, isolated Sami communities living in the southern parts of Finland even in the 18th century - pagans, no less.
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It would be pretty hard to devise ways their culture could change so much and their numbers grow in such a major fashion that they could expel the encroaching Swedes and Finns and forge a nation of their own.
Incidentally, I think the best way to get an actual Sami state, at least in name, would be to devise a plan to get the USSR to annex a part of the Finnish (and possibly Norwegian) Lapland in 1944-45 and to establish a Sami ASSR there as a Sami homeland under the fatherly guidance of the Soviet government. Maybe I'll write a TL about that at some point.