Answering a very old post here, but:
- Language issues:
Finding Sámi people who spoke Norwegian or Swedish as a second language was not difficult for Linnaeus, and indeed his diary mentions several instances of him being in direct communication with Sámi. Sámi people have had to relate to non-Sámi authorities and non-Sámi neighbors for a long time, so multilingualism is no new thing in Sápmi.
- The place where he wrote down the rules:
The last toponym mentioned before the "fun and games" section of the diary, where Linneaus presents Sámi games, is Lake Virihávrre, which he passed on his way back from the Folda Fiord in Denmark(-Norway).* However, he was likely in Kvikkjokk, which was under Swedish dominion, when he wrote down the rules: Before the rules are given, Linnaeus says that he has arrived at "hyttan", in English "the cabin." This could be interpreted literally as Linnaeus having arrived at a cabin, but it is much more likely to mean Kvikkjokk, which at the time was known simply as Hyttan** (capital "H"). This is Linnaeus' personal diary, in which he uses shorthand, there are broken-off sentences, the text shifts freely between Latin and Swedish etc. If we assume that "hyttan" is supposed to be "Hyttan", then the concrete place where the rules were written down has been located. Indeed, not long after the passage on Tablut Linnaeus describes the river at "Hyttan", and he then uses a capital "H".
That said, even if we have a pretty good idea about where he wrote down the rules (Virihávrre and Kvikkjokk are not very far from each other) we still don't know where exactly he learned the game. We only know the district: The Sámi words used are identifiable as Lule Sámi, meaning that he learned the game from Sámi somewhere in the Lule Sámi area, which is relatively large and where he visited several places.***
However, the section on games comes in the middle of a long description of different aspects of the culture of the reindeer-herding Sámi of the highlands that Linnaeus is travelling through at the time. In that context it is also of interest that the gameboard is described as embroidered on a reindeer skin. This way of keeping (and transporting) games is also known from a reindeer herding Sámi community elswhere in Sápmi (a Sáhkku board drawn on reindeer skin, where the skin doubled as a bag for the pieces during transporation).
It does not appear unlikely that the game was taught to Linneaus by reindeer-herding Sámi in the Folda-Kvikkjokk area, although one cannot entirely exclude the possibility that it was taught to him by farming or ocean-fishing Sámi in the same area. But if it was taught to him by the reindeer-herding Sámi, as is likely, then looking for a concrete village to connect Tablut with may not be altogether fitting, since the reindeer herding Sámi are semi-nomadic, travelling seasonally within their district instead of living fixed in one village. The game would have been brought along from inland to coast and back again, from the part of Sápmi claimed by Denmark to that claimed by Sweden**** and back again.
As such, the place of origin for this Tafl game is not possible to sum up more concretely than "the Folda-Kvikkjokk area, a part of the Lule Sámi area that transcends the current Norwegian-Swedish state border."
-----NOTES----
*Norway was at the time part of Denmark.
**Hyttan literally means "The Cabin", but in this case referred to a specific building used for silver smelting, which gave the place its name. The Sámi name of the village is still Huhttán, a borrowing from the old Swedish name - while the current Swedish name, Kvikkjokk, is a borrowing from the Sámi name of the local river Guojkkajåhkå (meaning "River of Rapids").
Incidentally, there are also other parts of Sámi gaming history connected to Kvikkjokk. The first documented use of the word "Sáhkku" to signify a board game is from here (in 1841, "sakkotet" is mentioned as a word for "playing a board game" in that area), and a dablo board from this area is preserved at a museum in Sweden.
***The Lule Sámi area covers the Salten district in Norway and a wide area around the Lule River in Sweden. It crosses the Scandinavian peninsula from the west to the east coast. Map:
https://en.wikipedia.org/wiki/Lule_Sami
****There were no fixed state borders between Denmark-(Norway) and Sweden through the Sámi highlands at the time of Linnaeus' travels. These would appear later in the 1700s. Linnaeus himself states at one point that he assumes he can now say that he's entered "Norwegian Lapland" because the rivers have started running westward instead of eastward. This watershed-based understanding of borders was of course not relevant for the reindeer herding Sámi in the same area, who freely traversed the mountains between the western coast and eastern forests, and their right to do so was respected for a long time after the Danish-Swedish border through their lands was set.