Judging by the LotR movies, and even by Legolas' actions in the books of The Lord of the Rings
You pretty much answer your own question here, but then take a wrong turn with:
Does anything in Tolkien's writing support the common notion that Elves are, first and foremost, archers by nature and inclination, and employ other weapons only at need?
In the Silmarillion elves use the same kinds of weapons as the humans: swords, spears, bows, axes. In fact, nothwithstanding the fact that they're the Firstborn and so on and so forth, elven culture & human culture in Beleriand is not highly differentiated. Posters above cite the examples of elven archers, but as noted these are hardly dominant.
However: the LOTR books give a very different picture. Elvish culture in the Third Age is not much like human culture at all, and Tolkien repeatedly emphasizes their skills at archery, the fact that they're fast and un- or lightly armored, and so on.
He primarily does this with the character of Legolas who, because he's the only elf Companion, stands in for all elves in many ways, just as Gimli becomes the quintessential dwarf. This depiction of elf-as-archer painted by Legolas is further reinforced by references to the Mirkwood elves, and by our meetings with the Galadhrim.
So in reference to Tolkien's actual work, among the elves of the First & Second Age the bow did not dominate, whereas among the elves of the Third Age it did. The posters making the points about the different types of elves who remained in Middle-earth by the Third Age have, I think, explained why.
In reference to readers' interpretations, however, the stereotype of the elf-as-archer is completely justified since that's overwhelmingly the way Tolkien portrays them in LOTR, and LOTR is where the vast majority of Tolkien readers are introduced to his work.
So the point "Apart from LOTR, does anything justify..." is moot: LOTR is the canonical work, and - when considering Tolkien's work as a whole - must be weighted accordingly.
As an aside: the bow-carrying elf ranger trope certainly doesn't predate Tolkien since he invented both "ranger trope" and "bow-carrying elf trope" as we know them today (along with "orc trope", "axe-carrying dwarf trope", "Evil Empire trope", "tree-man trope", etc).
Second aside: Celtic & Scandinavian folklore called flint arrowheads that were found in fields "elf-shot" and associated them with "the Little People" (ie elves). It's impossible that Professor Tolkien was unaware of this, although I leave it to Tolkien scholars as to whether or not there is any explicit evidence linking Tolkien's elves with this particular dimension of their folkloric namesakes.