If a Moon Druid is wildshaped into a Giant Spider, which has a web attack (which is classified by the MM as a ranged weapon attack), can he then use the Sharpshooter feat to gain a +10 to damage since it's a ranged weapon attack anyway?
No, they can't use Sharpshooter on their web attack
Sharpshooter's third benefit requires you to use a ranged weapon, not to make a ranged weapon attack.
Before you make an attack with a ranged weapon that you are proficient with, [...]
Although Web is classified as a ranged weapon attack, it is not a ranged weapon, so it is ineligible for the ability.
From Jeremy Crawford's tweet (emphasis mine)
I'll boil it down for you. When we wrote the "Player's Handbook," we meant the weapons on the weapon table when we wrote "weapon," unless we said otherwise. But you break nothing in the game if you let natural weapons go along for the ride.
and another one
Their attacks count as weapon attacks, but the system doesn't consider their body parts weapons.
Although he said you can let natural weapon be counted as weapon, remember that web doesn't deal any damage, so in line with "net cannot benefit from extra damage, like Sneak Attack and Sharpshooter", web also cannot benefit from extra damage too.
And another Crawford's tweet regarding this
The intent is that a net doesn't deal extra damage because it's not dealing damage in the first place.