There are two common types of surface webbers: brown recluse spiders and phantom spiders. Both are vermin creatures and will keep spinning webs if they exist on the map at all. Large cat populations can exterminate them, though.
Brown recluse spiders live in "temperate broadleaf forest". In savage forests of this type (joyous wilds, untamed wilds, terrifying), the modifications "giant brown recluse spider" and "brown recluse spider man" can show up. They're humanoid creatures resp. very large spiders and will only show up as random wandering monsters. If you don't capture them and employ them in a silk factory scheme, they'll show up, walk around a bit, leaving webs, and wander off after a few game weeks. For sustained web production without massive effort, you need the small vermin type.
Phantom spiders are only met in evil forests. According to the raws, any (evil) tropical or temperate forest. Being vermin, they should be immune to husking dust, but your web collectors won't be.
Webs on the surface are destroyed by any type of precipitation that hits them - ordinary rain as well as vile filth falling from the skies. In a very wet biome, webs may be washed away before your weavers can collect them.
I don't know how vermin populations are allocated to a site; it may be that the types you get are somewhat random.