Dtyjlui

The bullets are made from Adina cordifolia.

Adina cordifolia aka Haldina is a decideous tree that grows up to 20 meters (65 feet) high.

Haldina cordifolia, syn. Adina cordifolia, is a flowering plant in
the family Rubiaceae, the sole species in the genus Haldina. It is
native to southern Asia, from India in the east to Yunnan and Vietnam
and south to Peninsular Malaysia. It is known as Kadam or Kadamba in
Hindi and Gáo trò in Vietnamese. (ref)

It is used to make furniture and flooring.

Haldina is part of the Rubiaceae botanical family. The same family coffee is from.

Why would this wood work when metal fails?

For the same reason that silver bullets kill werwolves when lead bullets fail. Or why a small piece of crystal from their home planet will take away the powers of Supergirl and Superman when bullets, bombs, or being hit by a train does nothing.

In other words, it works because you say it works. Make it part of the allergy. Even brief proximity to anything related to coffee will weaken this monster.

The wooden bullet is strong enough to hurt a human...seriously, who wants to be hit by a small pointy stick going 950 miles per hour (427 meters per second, the speed of a slower bullet)? That would kill any human if it hit the right place.

Because the bullet is "made of coffee" (close enough for your purposes), it weakens the monster's body enough to allow the bullet to penetrate. The speed and force of the bullet does the rest.

EDIT: Why not just use a wooden stake?

A stake would work if you could get close enough without dying first. But this monster can sense the presence of coffee from a distance farther than its reach (since surrounding yourself with coffee grains works to keep yourself safe). It would know you were sneaking up on it with a stake and would move out of reach.

With a bullet, you can hit the monster from a distance. And with enough force to do some damage, even if you don't hit its soft spot.

Only the locals know the secret

The invaders with their huge elephant guns can blast away for as long as they like. What they don't know, and the local tribes long ago discovered, is that the armour is actually fibrous and the fibres lie at right angles to the monster's skin (rather like a coir doormat).

These fibres easily mat together temporarily to block a bullet and then bounce back to shape, but if the monster is shot with a needle-sharp, wooden arrow or dart, it parts the fibres easily and penetrates right through. A simple poison tip then finishes the job.

Why does a wooden stake through the heart or sunlight kill a vampire and bullets, knives, and other death dealing implements don't? It's part of the mystique of the monster.

Maybe the skin of the monster "senses" a metal bullet and turns into armor for the impact, then softens again. Sort of like oobleck, but better, since Mythbusters tried to stop a bullet with cornstarch and water and failed. The skin can't detect the wood, so doesn't harden and the wood is able to pierce the skin into the vital organs.

It all comes down to what you want to do. It's your book, you don't have to stick science. You already have the mythical creatures tag, so it, pretty much by definition, doesn't have to make sense.

Can someone please help me with an explanation about how a piece of specific wood could pierce through armor even when other harder, stronger materials can't?

You're going to have to improve the armour before the question makes sense, because e.g. kevlar bullet-proof armour is famously not stab-proof. Kevlar stops fast relatively blunt objects, but not slow sharp blades, so being bulletproof does not inherently protect against metal blades.

Maybe you could come up with another reason why metal blades are no use (perhaps a dense acidic sweat which corrodes steel unreasonably fast, blunting the edge?) and make ironwood spears the weapon of choice against your foe.

Perhaps the armour is a bit like a non-newtonian fluid (ok, solid in this case) that doesn't only react to force but also reacts/hardens depending on the density of the thing hitting it.

My balsa wood arrow doesn't have the density or velocity required to make it react and slips right through, but your dense fast-moving bullets make it tougher than Kevlar. Unfortunately, balsa is too soft to make a good arrow, but over the years we've found that handwavium wood is low density and really strong across the grain and that's why we use it against these monsters.

Since it's natural armor, I imagine it's made of some kind of keratin. Maybe reinforced with fiber.

Without getting into the specifics of the wood or the keratin, let me suggest that some quality of the of the sap in the wood is like silver nitrate solution (AgNO3) and some quality of the keratin "glue" is like salt (NaCl).

The reaction moves very quickly (see this video) and moves even more quickly in cold environments (like normal temperatures).

So again speculating with this hypothetical pair of reactive sap and keratin, lets say the output products are significantly weaker than the undamaged armor. Additionally, the chemicals that result from the reaction may cause a secondary reaction with the fibers reinforcing the armor, causing those polymers/fibers to curdle (see video).

The compound effect is that a simple touch from the wood causes the natural armor to gap open, and this happens quickly enough that a slow-moving attempt to penetrate the armor (such as stabbing it with a stake or spear made of this wood) just seems to pass through with inexplicable ease.

I recommend quebracho, a wood so hard that it supposedly breaks axes.

This creature has a pseudo magical outer shell of unknown molecular structure. This gives it strength... But a specific wood type also has a molecular structure that loves to chemically interact with the shell of this creature. This interaction quickly weakens the surrounding structure and allows you to pierce through.

The answer is even easier than you think.

Each wooden bullet would have to be made from the 'lodgepole' or core of the tree. As the tree grows layers develop at higher levels on this core. The bullet, looking at it end on, would have rings. When fired the bullet, conical in shape (7.62 NATO shape) would shed layers as it passed through armour. It would act like a depleted uranium round, heating up and hardening as it went through. Velocity becomes yur friend. Copper sleeve the round for stability, which would shed on the armour.