Yes. Conjure Volley will damage lycanthropes.

Here's the relevant part of the Conjure Volley spell description:

A creature takes 8d8 damage on a failed save, or half as much damage on a successful one. The damage type is the same as that of the ammunition or weapon.

The weapons don't do the damage, the spell does.

Here are the immunities for a werewolf (SRD, p. 331):

Damage Immunities bludgeoning, piercing, and slashing from nonmagical attacks not made with silvered weapons.

Conjure Volley isn't a 'nonmagical attack'. It's not an attack of any kind: it's a spell that does damage.

Contrast with a black pudding, which is just immune to slashing damage. Conjure Volley using (say) handaxes would do zero damage to a black pudding, but full damage to a werewolf.

Conjure Barrage is not a ranged weapon attack.

This does not work because "ranged weapon attack" in D&D 5e is a term of art--it refers to a specific piece of the rules, and not necessarily the underlying fiction. While the effect of conjure barrage may intuitively seem like a ranged weapon attack, it is not a "ranged weapon attack" as far as the rules are considered.

A "ranged weapon attack" is partially defined on PHB 194:

When you make an attack, your attack roll determines whether the attack hits or misses. To make an attack roll, roll a d20 and add the appropriate modifiers.

On the other hand, conjure barrage asks for a saving throw—it is the target that rolls the dice in that situation. Because it calls for a saving throw, and you don't make the attack, it doesn't work.

Compare this situation to a spell like fire bolt, which asks you to make a ranged spell attack. In this case, you're making an attack, so you roll the d20, but it's still not a "ranged weapon attack" because it's a "ranged spell attack".

Essentially, you need to fulfill all three words of the phrase. You must make an attack (you roll the attack roll), at range, using a weapon.

Note that you can't cast those spells in one turn.

The wording in your question is a bit ambiguous, but it's important to note that you can't cast both of those spells in one turn, because that would require a bonus action spell and a 3rd level spell. PHB 202 states, (emphasis added)

A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven’t already taken a bonus action this turn. You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.