Mantis Shrimp Claws will do quite nicely.
Carbon is extremely abundant on Earth's surface and presumably any other Earth-like planet that the author may be working on. Given the many forms that carbon can take from the ultra-soft graphite to the ultra-hard diamond, it should be able to satisfy your needs.
Characteristics of good armor
The element(s) that form this armor are important but the construction/organization of the elements are far more important. If the armor is too rigid, it will shatter. If it's not hard enough, then the bullet will pass through. (An example of a behavior we don't want is spider silk. True, it's stronger than steel at that scale but it's also super stretchy. Stopping a bullet on the other side of the target isn't very useful.) The goal will be to spread the bullet's kinetic energy over a large enough period that the armor plate can handle it.
Mantis shrimp have ridiculously resilient armor. They have to since they hit harder than most anything else in the animal kingdom. Note the many layers of fibers at slight angles to each other. In this configuration, penetrations that break through between two parallel fibers (this is the weakest configuration) runs into the next layer below that is oriented more towards the longitudinal orientation (which is far far stronger). At every layer, the bullet is forced to expend lots of energy breaking the bonds of the fibers along their strongest axis.
Fast Replacement Armor
Requirement 4 is the most interesting. Growing plates like a turtle shell would certainly be effective from an armor perspective however, these don't grow quickly. Human skin doesn't offer any kind of armor capabilities but it does grow very quickly. Skin wounds can heal in a month or less (depending on various factors). Clearly we need something that will grow fast and ideally is always growing. If the armor is damaged, we don't want to have to keep carrying it around for longer than we have to. Lots of animals have disposable "armor" to one degree or another. Humans have their skin. Porcupines have their quills (which are replaced). Snakes, lizards, crabs and lobsters have their skins.
Let's assume this creature is a carnivore so that it can afford the higher metabolic costs of replacing all it's armor in a month or so. Perhaps as a way to reduce this metabolic load, the creature swallows the old armor scales which are then broken down into basic components for the armor-building cells to use.
Alternatively, the outer layers can just flake off after exposure to oxygen for some period. This gives the armor a natural decay rate and prevents the armor from getting too thick. Natural variation in the armor breakdown proteins could lead to some creatures with thicker or thinner armor than others. Hey cool! We just invented a way to get heavy and light versions of the same creature suited for different battlefield duties without having to breed different versions. Win!
Yeah, but how good is it?
Mantis shrimp 'fists' are known to withstand 4 gigapascals. This is about 40k bar or 1/90th the pressures in Earth's core. Dang.
I'm going to assume a NATO 5.56x45mm round. It's super common and well understood. With a muzzle velocity of 990m/s. Kinetic Energy is:
$$KE=\frac{1}{2}mv^2$$
$$\Delta E=F\parallel d$$
therefore
$$F=\frac{\Delta E}{d}$$
Assume $E_0=0\text{ J}$
therefore $F=\frac{mv^2}{2d}$.
$$P=\frac{F}{A}$$
$$A=\pi r^2$$
therefore $P=\frac{mv^2}{2d\pi r^2}$,
where $P$ is the pressure in Pascals, $m$ is the mass of the bullet in kilograms, $v$ is the muzzle velocity in meters per second, $d$ is the distance the bullet travels in meters, and $r$ is the radius of the bullet. With all that, we get that the pressure exerted on the armor is
$$\frac{.004\cdot990^2}{2\cdot d\cdot\pi\cdot0.00285^2}\approx \frac{77}{d}\text{ Megapascals (MPa)}$$
The farther away the gun, the less pressure exerted. With simple algebra, we can find that you would have to fire from just under 2 cm away to reach a breaking pressure of 4 GPa.
This is only an approximation since these calculations don't include angle of impact, thickness of armor, ablation effects, the liquid characteristics of metals at high speeds and small time frames, tungsten at high speeds, possible pyrophoric effects, and so on.
Turn it up to 11
So far we've been talking about common mantis shrimp armor. Cool. Let's turn it up to 11 by replacing whatever carbon/calcium materials are in their fists for carbon nanotubes. Given that the theoretical maximum for carbon nanotubes is approximately 100gpa (about 25x our baseline), replacing a substantial portion of the default fist matrix should yield impressive strength gains. I'm no materials engineer so I can't prove it. I only play one on the internet.