In my conworld, the inhabitants have futuristic technology and use armour that is made of primarily wood but will absorb bullet impacts and even armour-piercing rounds. It is cited as being very fire-resistant, though after enough time has elapsed it will burn through. What would be the conditions required to make wood this durable? Keep in mind, the planet these trees grow on is 1.5 times as massive as Earth, and the atmosphere is much richer in oxygen.
This question made me smile, because I think there might actually be a tree on Earth that would stop a (lowish) caliber bullet.
I cut down a Black Gum tree (Nyssa sylvatica) that was shading too much of a garden (no great loss; my land had thousands of trees.) I cut in into lengths to split for firewood. It was dense and somewhat harder to cut than most other trees/wood.
But the real surprise came when I tried to split it. It completely overwhelmed my heavy duty hydraulic splitter. The grain was so twisted every which way that I had to just give up and haul the lengths into the woods.
Because of its twisted grain, black gum cannot be split so it has been somewhat spared by lumbermen. The resilient wood is good for making tool handles.
The wood of the black gum has fibers not only interwoven but twisted to boot. And while modern tools and cutting edges can surmount this obstacle, even the sharpest ax and the most expertly wielded froe [a tool for cleaving wood] of yesteryear made little headway splitting it.
Is an ax very much different from a bullet in shape?
Consider a tree that has highly twisting and interlocking fibers. A bullet can make some headway, but nothing close to what it can do to most woods with straighter grains.
If early settlers left it alone, it had to be formidable. Make your tree a bit more so.
In order for this to work, the composition of your trees would have to be VERY different from those on earth. Even dense wood is not particularly good at stopping bullets. A handgun round will generally penetrate wooden boards as much as 2-3 inches thick, and high-velocity armor piercing rounds from sniper rifles will go through as much as a foot or more of oak.
Those kind of thicknesses are pretty impractical for body armor, so you're going to need something pretty different.
Now, something that DOES grow naturally that IS bulletproof, or nearly so, comes from the animal, not vegetable kingdom. Some species of Mantis Shrimp form structures in their claws that have incredible levels of shock resistance, and would easily generate both the protective levels you're looking for as well as being practical for use as armor.
Instead of trees like we have on earth, imagine something much more like a coral forest where the corals form these kind of structures as they grow. Your inhabitants can harvest them and repurpose them as protection. There's all sorts of potential there.
There are examples of biological systems depositing glass (diatom skeletons, for instance). Glass fibers encased in lignin in between the cellulose fibers would greatly reinforce a high-density wood (like teak or boxwood). The result wouldn't be "bullet proof" any more than modern body armor is "bulletproof" -- but it would be bullet resistant in much the same way. Layered scales of this wood, with padding between layers (made from the self-reinforced bark of the same tree?) could gradually absorb the energy of a bullet.
The tree that would have evolved to do this would have to grow in a silica-rich region (sandy soil, at a minimum) and would have an evolutionary need to grow tall, yet stand against strong winds. Think solitary redwood or sequoia -- instead of being protected by standing in a grove of similar trees, these are simply strong enough to keep standing and hold their branches in the most violent storm. The slow growth and long life cycle of these trees would, however, make the armor made from their wood relatively rare and expensive (especially after sustainability becomes a recognized concept).
As an actual wood by itself with no special preparations, almost certainly not, cellulose is just too weak to do this by itself if we assume equivalent structure to real-world trees. Even the highest density woods (which are so dense they sink in water) aren't going to get you close to a proper anti-ballistic material like modern body armor provides.
However, if you do some form of multi-layer laminate, you might get acceptable performance against ball ammo (that is, ammo that doesn't have an armor piercing core). Essentially, you're making armor out of plywood, except you want a whole lot more layers, and probably want each layer thinner than typical plywood (you can pack more in, and better see any imperfections in the grain of each layer during assembly that would lead to weak points). You'll probably want hardwoods for this (more durable), but most likely softer hardwoods (if they're too dense, they won't do a good job of absorbing the kinetic energy of the bullet). Even then though, you're looking at maybe stopping handgun rounds and some light hunting rounds, probably not military-grade AP rounds, and definitely not anything larger than about 8mm coming from a rifle.
The simple answer is "No".
Wood is based on cellulose, and cellulose has very low tensile strength, compared to "strong" materials like kevlar or steel. Cellulose-based materials just can't be "bulletproof" in a traditional sense (ex. to make wearable armor).
However, plants in your world can potentially be made of other fibers. You can even have some kevlar-based vines, if you like.