The Key is how your GM handles searching for traps.
If your GM allows perception for checking for traps:
Then elf Keen Senses gets you proficiency with perception and sub races to get proficiency with a ranged weapon:
Drow: Drow Weapon Training = Hand crossbow
Sea Elf: Sea Elf Training = Light crossbow
Eladrin (Variant), High Elf, and Wood Elf: Elf Weapon Training = shortbow and longbow
Elves of Valenar: Valenar Elf = shortbow and longbow
Human (Variant) - Select perception as the skill and feat Weapon Master Hand Crossbow, Heavy Crossbow, Shortbow, and longbow
If your GM uses investigation for checking for traps:
Human (Variant) - Select investigation as the skill and feat Weapon Master Hand Crossbow, Heavy Crossbow, Shortbow, and longbow
Then to be sneaky you can use:
The urchin (PHB) and Urban Bounty Hunter (Sword Coast Adventurer's Guide) will give you stealth for the sneaky part.
Then you just need a class that is functional with ranged weapons. …. But that often becomes redundant.
We don't care about heavy armor or shields because you need to hands and your going dexterity.
Barbarian = Light + Medium Armor, Simple + Martial - No relevant Archery ability I can think of
Bard = Light Armor, Simple (Light Crossbow, Shortbow) + Hand Crossbows - No relevant Archery ability I can think of
Cleric = Light + Medium Armor, Simple + Martial - No relevant Archery ability I can think of
Druids = Light armor - No relevant Archery ability I can think of
Fighter = Light + Medium Armor, Simple (Light Crossbow, Shortbow) + Martial (Hand Crossbow, Heavy Crossbow, & longbow) and allows for archery fighting style as well as then enhanced extra attack.
Monk = Simple (Light Crossbow, Shortbow), Unarmored Defenese voids the need for armor , Kensei lets you use Light Crossbows, Shortbows, longbows, hand crossbows and blowguns as a monk weapon benefiting from martial arts die.
Paladin = Light + Medium Armor, Simple + Martial - No relevant Archery ability I can think of
Ranger = Light + Medium Armor, Simple + Martial , archery fighting style + Archer magic!!! If your primary goal is archery <-This.
Rogue = Light , Simple + hand crossbows, expertise and reliable talent if you want to be board and waste the tables time and annoy your GM by not being able to fail to spot or disarm traps with DC27 or less... (kind of boring if you aske me). … oh and backstab incase you want to do way too much damage that is almost unrelated to your attack, if you don't believe me attack a second time.
Hexblade Warlocks (ignore the rest) = Light + Medium Armor, Simple + Martial -Eldritch smite with a bow... worth it? eh...
Sorcerers and Wizards have Darts, daggers, and slings... I mean it not archery technically but you could... I mean I would … but normal people are going to say no.
Typical Waste Example: A Drow Rogue can get expertise with perception and stealth (+investigation and either athletics or slight of hand), thieves tool proficancy and both rogues and Drow have Hand Crossbow Proficiancy… so super redundant. It works but its very typical Drow. If you want to play that go for it.
More efficient slightly more interesting build: Wood Elf rogue gets a longbow which is not a normal rogue weapon. They are likely to be mistaken as rangers but still have perception (elf), investigation, stealth, slight of hand, and maybe actobatics + thievies tools for traps. Then something like the soldier background (athletics and intimidation) for some interesting "why do have these skills" background.
Overly optimized build: Human variant Rogue with survival and crossbow expert, hand crossbow as a main weapon lets you attack again as a bonus action for another chance at backstab if you miss or use cunning action if you hit. Rogue skills investigation, perception, slight of hand, stealth and thieves skills. Backgound soldier for the same reasons.
More interesting build to defy expectation: Bugbear (+2 str, +1 Dex, Darkvison, stealth proficiency, and +2d6 on surprise hit) Kensei Monk with a blow gun and Acrobatics and Athletics. Take Uthgardt Tribe Member background for Thieves tools, survival skill, and since you already have athletics you are allowed to change if for one other skill of your choice and you take perception or investigation... which ever your gm uses for traps. At level 4, take prodigy for expertise in your trap finding skill and the other skill. Then your set. If your not sure you can succeed in a disarming a trap, just throw things at it, pull on it with robes around the corner and block it with the party tanks shield.
Sure you could go Fighter or Ranger with urchin instead... maybe even Warlock Pact of the blade enhanced pact weapon, but does really have the bow before level 3, until you change one of your invocations (unless you grab race bonus as above). The Bugbear monk can fire a shortbow at level 1-2 then switch to blowgun at level 3. I mean who wouldn't want to play a bugbear using a blowgun with extra attack and monk weapon damage of 1d6+dex+1d4 (Kensei’s Shot) + 1d6 Deft Strike (with use of Ki) x2 each maybe a little poison and a 2d6 ambush bonus from bugbear?!??!
