The Archer merit badge is fairly straightforward. You can even find a static plan for the Crossbow right at the scout camp itself, at, you guessed it, the archery range. Once you fast travel to the camp, you’ll go past the first sign and turn right, and keep going straight and you’ll be at the archery range (as illustrated in the linked video). If it’s not there, just try coming back again later, or on a different server. I imagine the plan is probably in demand at the moment, so it may take a bit of patience.
The full requirements for the Archer merit badge are as follows:
Pass “Archer” knowledge exam (on the computer at Camp Lewis)
Complete the Pioneer Scouts shooting test – I did this on my first try, on my first five shots, the first time I’d ever used a crossbow in the game. I’m not a fantastic shot, by any means, so if I can do it, you can do it.
Craft Crossbow Bolts (0/50) – Don’t forget the Ammosmith perk (under agility) and the Super Duper perk (under luck). Crafting three batches should be enough for this part, and probably the kills section, but it’ll most likely take quite a few more to wear out your crossbow enough for the repair requirement.
Repair a Crossbow (0/3) – Sorry. Just…we’re just sorry.
Kill enemies with a Crossbow (0/50) – Might take a little while, but it’s easy enough. And as a bonus, looting your downed enemies will give you back crossbow bolts often enough, so, yay?
So basically, once you either craft, find, or buy a crossbow, and pass the exam, you just have to do your best William Tell impersonation, and shoot a bunch of stuff. You’ll probably want to shoot a radstag, since that’ll also count toward your hunter badge, but don’t stop there. Targets, enemies, and anchovies, and orangutans, and breakfast cereals…oh, wait, that’s grenades. Sorry. Anyway, the most difficult part of this is likely wearing out a crossbow enough to repair it three times. Not using Weapon Artisan and Gunsmith will help. Also, once you repair it once, you should be able to just shoot it enough to get beyond the basic threshold to repair it again.