A Ranger casts the spell Conjure Barrage:

You throw a nonmagical weapon or fire a piece of nonmagical ammunition
into the air to create a cone of identical weapons that shoot forward
and then disappear. Each creature in a 60-foot cone must succeed on a
Dexterity saving throw. A creature takes 3d8 damage on a failed save,
or half as much damage on a successful one. The damage type is the
same as that of the weapon or ammunition used as a component.

For the component, they use a piece of ammunition that has been coated with poison - let's say wyvern poison:

Wyvern Poison (Injury). This poison must be harvested from a dead or
incapacitated wyvern. A creature subjected to this poison must make a
DC 15 Constitution saving throw, taking 24 (7d6) poison damage on a
failed save, or half as much damage on a successful one.

Does the target of the spell also incur poison damage from the poison? I am inclined to say yes, since it says, "The damage type is the same as that of the weapon or ammunition used."

Conjured ammunitions are not poisonous

The poison is not part of the ammunition.

The damage is not part of the weapon damage. "Make a save" damage never was. For instance, it is excluded from critical additional damage.

It is not cloned and therefore only deals the ammunition 3d8 piercing(?) damage.

Only the coated ammunition is poisonous, and since it is not shot forward by conjure barrage, none of the affected targets need to make the poison save.

However, this is a clever use of the spell, and since poison is usually underused in my session, I might allow this for one target failing their save (as if the coated ammunition is included in the barrage). Other DM might find this combo cool and subject all the targets to the save.

If a table use poison more often and you want to allow the combo, be careful with adding the poison on all targets. The additional damage is very powerful, even using lesser potent poison, that you could make one spell overpowered. They might create a lot of poison, then they could roll through your encounters and make one character outshine the others. Usually I made it a point on session 0 that overusing a cheesy strat would cause the enemy use the cheese against them, as a general rule. When I first rule this, I will explicitly say, "Okay, I'll allow this cheesy strat," and emphasize on cheesy.

Slightly related, Jeremy Crawford's unofficially tweeted:

A magic item like the flame tongue and the frost brand deal extra damage of a particular type when they hit. That extra damage doesn't change the underlying damage type of the weapon. And it's the weapon's damage type that is used by a feature like Sneak Attack.

So, you can't choose poison as the barrage's damage.

No

Spells do only what they say they do. Conjure Barrage deals 3d8 damage without taking into account size, damage die or other special properties of the used weapon. The spell does not say that it delivers any enhancements to the weapon damage. The only thing you can control is damage type.