I feel like most of the other races will use any old weapon if it helps them kill each other, to be honest. Besides the Bosmers' cultural wood allergy, I can't actually think of any lore that prescribes certain types of weaponry to certain races, at least not in a general sense (excluding particular types of weapon unique to those races, like the Redguards and their scimitars for example). Different races have a proclivity towards certain types of combat, but even this is just a generalisation. My thoughts are:
Imperials are known for their discipline and Roman-eque military tactics, so they strike me very much as longbow and sword-and-board users, particularly the Legion with its mixed units. Longbows are my go-to thought for the Imperials, as mentioned, but like the Romans who inspired them, the Imperials will adopt whatever manner of warfare works, and they aren't afraid to learn from other cultures and steal their ideas. Basically, the Imperials strike me as the ultimate 'jack of all trades' race. I do imagine that Bethesda meant the Imperials to be analogous to 'normal humans' really, like a default. They come in all shapes and sizes, fill all kinds of roles and vocations, and use all kinds of skills and equipment. This is true of all races to an extent, obviously, but the Imperials truly are 'generic', for lack of a better word.
I'd say the Nords are similar to the Imperials overall in terms of equipment and combat (Tiper Septim was a Nord, after all), though with an obvious preference for two-handed weapons. They are less disciplined and cosmopolitan, as a rule, but they are still effective warriors that have historically been successful in most forms of combat. That would include archery, as all successful armies would make use of bows. We see a number of Nord marksmen in Skyrim and elsewhere, using longbows and others. In fact, given their size and strength compared to the other races, a Nord may well excel at using the longbow, and might find it far more mobile than the other races to boot. They make up a large percentage of the Imperial Legion too, so they would have to become accustomed to their equipment and tactics.
The Dunmer strike me as battlemages, first and foremost, with a focus on destruction magic for range. Having said that, there's nothing stopping them from using bows, and indeed we have seen them do so, both in Morrowind and every other game in the series. Again, when it comes to war, people will tend to use what works, and we know that bows work, both in the real world and in Tamriel. The Dunmer are smart and shrewd, and wouldn't eschew the use of longbows for any particular reason.
We know the orcs use longbows, as we see them do so. Orc hunters in Skyrim are equipped with longbows, and they use them for hunting. And like the Nords, the Orcs are important constituents of the Imperial Legion, so many of them would have learned the skill from there, and their race would have no issues with using them. They do have a reputation as brutes favouring melee, but this reputation doesn't really stop them from using anything they put their green hands on.
Argonians are obviously bow users too, as are Khajit, but each has a racial preference for certain types of combat (unarmed, in the case of the Khajit, from what I remember of the lore). Here, too, we have seen both races use bows of varying types, with varying degrees of skill. And certainly, given their penchant for stealth, a bow of some type - though perhaps not a longbow, in the traditional sense of the word - would be useful. The Argonians and the Khajiit have seen their fair share of war, both as independent forces and as part of the Empire, so they'd know the value of a longbow in wartime too. I don't know if any lore has gone into particular detail about how those wars were fought, though, at least not to the point of describing their armaments. But generally speaking, both races would probably prefer mobility over power and range, so the longbow might not suit them hugely. And I can't imagine the Argonians fighting with longbows in a swamp, soo...
The Redguards, like the Imperials and the Nords, will seemingly use whatever helps them to win wars, and bows would likely be a significant boon to their guerilla warfare. I would imagine composite bows are important to them owing to their mobility, though, as you suggested.
The only outliers so far as I can tell, really, are the Altmer and the Bretons.
The latter certainly make use of bows, but I wouldn't really identify them with such. Not all Bretons are magic users, and even the Forsworn (I know, I know, they don't like to be called Bretons, but they typically are...) use bows alongside their magic and melee.
The high elves are notoriously magic users, so they are the only race I can really think of that would cause me to do a double-take at first glance if I saw one wielding a bow. Having said that, they do have their own distinct form of bow - the elven bow, which I think is meant to be like a recurve bow - so archery certainly isn't an alien concept to them, and we see the Thalmor soldiers using them in Skyrim. Like the Nords, they're rather tall, so may find longbows less unwieldy than the other races (though I don't know how strong they are compared to the other races, given how lithe they are). But given that they have spawned their own type of bow, one that we know to be highly prized across all Tamriel (the master archery trainer in Oblivion wanted you to get one for them, if I remember rightly), I would suggest that high elves would probably be best limited to their own type of bow, with the proviso that there is nothing really stopping them from using any other kind besides their racial superiority complex ("I'm not using that inferior bow that an Imperial made! I'm going to use a master bow, made by a master race! Mwuhuhahaha!").
The Bosmer are the only ones I can really think of that are particularly limited by lore as to what kinds of bows they use, which is ironic as they invented the bloody things.
So really, anybody but a Bosmer, and perhaps an Altmer depending on your point of view, would be OK using a longbow!