Jeremy Crawford ruled about the use of nets as illegal for the conjure barrage spell. So, not allowed under the Rules as the Designers intended for a character to be allowed to do so.
Conjure volley is meant to work with ammunition or a weapon that deals damage.
This follows logically with the text of the spell in regards to the type of damage dealt. Nets don't deal damage, so they have no damage type, and the spell clearly reads:
The damage type is the same as that of the weapon or ammunition used as a component.
And, before you ask, Jeremy Crawford already ruled, all damage has a type:
All damage has a type. #DnD
So there could be no damage done, as there would be no source of the extra damage.
The spell also says that the copies don't remain.
cone of identical weapons that shoot forward and then disappear
So, even if it worked, the second it hit the whole point of using nets would be lost, as they would immediately vanish.