In the books (not in the film) between Shire and Bree, Frodo and hobbit fellows met (before or after Tom Bombadil I can't remember) a group of Elves. Their leader told them he could try to offer protection from Nazgûl, if he would have a battalion of elven archers like ancient times.
According to The Silmarillion, Elves lived in open fields, in strong castle cities as well as forests. Tolkien does not classify them specifically as "wood elves", neither tells that Elves always lived in the forests. But at the time LotR takes place, Elves are weak in numbers and civilization, and therefore they prefer hidden in the woods.
But there are enough number of references in the books to make us sense that bow is a preferred standard weapon for Elves, no matter where they used to live.
Immortality is related to that preference not only for personal choice. As an immortal race, Elves also are rare, and most of times been, outnumbered, and strategically more limited in "casualty rates" therefore it is more costly for them to lose their members. A successful use of ranged weapons in correct combination, is the safest way to minimize casualties, given that you have much better archers in mass scale.
Any member of any race, who can master bow skill, would go after it. (Humans and Orcs have also specialized ranged units) However, Elves as a society, have more reason, more ecological pressure, and have more resources (time, material, skill, etc.) to make bows standard combat weapon, no matter how good they are at other weapons.
I don't share the opinion that Elves' "small size" is a point. Longbows require pulling strength, and archers in real history, are certainly big, tall, strong men, who were also very strong opponents in close combat, often deadly with a crude melee weapon, a dagger, an axe or hammer. Close combat specialization does not require "more body and muscle mass" than archery, but simply different tactics, equipment and skills.
In real history, bows are really most skil-dependent weapons. Actually, the reason why bows disappeared from battlefield, even before modern rifles became so accurate and deadly, is the problem of training time. You may train a bunch of peasants for a few months to have a standard unit of musketeers. An archer who can use a long bow (composite or non composite) effectively and safely in battle, requires a training of 10-12 years at least, and a "sea archer" or a "mounted archer" cannot become even good enough with the weapon before 20-25 years. Lost musketeers can easily be replaced, whereas, lost archers cannot. (Ottoman archers were expected to "pull the string and set the bow" to develop strength and understanding of the limits of the bow, string, and their arms. They were not allowed even to shoot an arrow before they advanced a few years in their initial training.)
