For single shot "zip gun" type weapons, almost any material can do. Indeed, with the restrictions given, it is difficult to imagine a firearm in the sense that we know today as being possible.
To make a modern "repeating" firearm, the challenge is multifold. The chamber needs to contain high pressures and high temperature gasses. The cartridge case is made of brass in modern weapons since it can expand slightly during firing to create a seal in the chamber, and its ejection helps remove a lot of heat from the chamber and weapon. It also helps that the metallic cartridge also mechanically protects the round and propellant in storage and in the magazine as well. Polymer cartridges have been experimented with (Notably the LSAT program), and caseless cartridges have also been developed (notably for the HK G-11 rifle program), but with limited success to date.
LSAT cased telescoped polymer ammunition as developed for a machine-gun
HK G-11 with 50 round magazine and caseless ammunition
The next challenge is to have a barrel which can handle the severe forces imparted by having a high speed projectile passing through it, along with heat buildup and mechanical strength to deal with soldiers bashing it in the field. Ceramic barrels might have some of the properties desired, but ceramic/metal composites seem to have the sorts of material properties to make an "ideal" barrel. Lacking metals, a ceramic barrel with a high strength material "wound" around it should have most of the properties desired, at the expense of a complex manufacturing process. The Carl Gustave M-3 has this on a larger scale in an 84mm recoilless rifle, but the rifled liner is still steel.
So the short version is firearms the way we understand them are going to be exceedingly difficult to make without metal. There are some possible substitutes, however.
Rocket projectiles have been around since ancient China. Using cardboard tubes filled with gunpowder attached to an arrow provides a simple yet effective projectile weapon, which can be fired from a crossbow like weapon individually or from a multiple rocket launcher to bombard an area with arrows
A more modern variation was the Gyrojet rocket pistol and carbine, using a .45 cal projectile powered by a tiny solid filled rocket motor. Oddly enough it was quite safe at point blank range (you could stop the projectile by putting your thumb over the barrel), but became increasingly dangerous as the rocket built up velocity. Accuracy was ensured by having the rocket exhausts angles slightly to impart a rotation on the projectile as it was in flight.
Gyrojet Mk 2 pistol
Gyrojet rocket carbines
The pressures inside the weapons are much lower than a conventional firearm, and they can be made of virtually any material (the historic ones used aluminum, but in principle even a plastic barrel would do). Polymer and ceramic could conceivably make a modern Gyrojet, should anyone be interested in reviving this, or for the purposes of your story set in the "modern" age.
Edit to add: https://www.youtube.com/watch?v=VdW8Trh_MGg
This video shows a successful firing of a Mk2 pistol, and you can see the rocket spinning the round.