As an edge case, the RAW technically allows this

You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of Ammunition. [...] A creature hit by the Poisoned weapon or Ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage.

The problem here is that this is an edge case. Normally, a slashing or piercing weapon is a weapon that does only slashing or piercing damage. Nowhere in the rules does it explicitly say that the poison must be delivered by those damage types. But since the rule only mentions the weapon and a halberd is a weapon that does slashing damage, technically the poison could be applied here even though the Polearm Master attack itself only does bludgeoning damage.

Logically, this shouldn't work

However, if we look at the rules from the perspective of what they are trying to say instead of the strict letter of the rules, I think we come to a much more reasonable ruling.

The rules state that a slashing or piercing weapon is what needed to deliver the poison. In the rules, damage types are almost always attached to weapon damage and not weapon identity. So, this wording is very unusual. Even so, the damage type is clearly a reference to the expected damage type of the weapon damage. Thus, I think it is highly reasonable to read the poison rules as saying "slashing or piercing damage from a weapon" because of this.

In fact, one of the sample poisons from the DMG (published after the PHB where the Basic Poison is written) is already written this way¹:

A creature that takes piercing or slashing damage from an object coated with the poison is exposed to its effects.

Narratively this all makes sense as logical inference from the rules seems to be that the target must be cut or pierced in order for the poison to enter the body and take effect.

^{1 - Thanks @NautArch}

There are no numbers given for how much poison is contained in a vial of basic poison, how big that vial is, or how much a single "dose" of poison is so it's hard to say with 100% certainty and ultimately comes down to DM ruling.

Option 1

A vial from the equipment list can hold up to 4 ounces of liquid (PHB, pg. 153).

Assuming a vial of basic poison contains a full 4 ounces of poison, then this gives us 25% weapon coverage per ounce of poison (or 12.5% per half-ounce), and 75% coverage on 1 piece of ammunition per ounce (or 37.5% cover on 1 piece of ammunition per half-ounce).

In short, you wouldn't be able to cover even one piece of ammunition, let alone one weapon, with the half-ounce of poison the alchemy jug produces and it would take you at least 8 days to produce enough poison to fill a vial.

Option 2

However, page 139 of the DMG states that:

Most potions consist of one ounce of liquid.

Basic poison isn't a magic item or a potion but it is clear that a 4 ounce vial isn't the standard liquid container.

If a vial of basic poison is only one ounce then you could cover half a weapon or 1.5 pieces of ammunition and would only take 2 days to gain enough poison for full coverage.

Option 3

Option 3 is to keep it simple. Don't get bogged down with unnecessary levels of finicky detail trying to figure out how many ounces are in a single dose of poison; the designers sure didn't in this instance.

If we keep things simple then the half-ounce of poison produced by an alchemy jug is equivalent to and follows the same rules as the vial of basic poison.