Several come to mind:
No, not the modern, short-fiber crap you get from the dollar store. You want the good stuff made of long fibers like flax or cotton. (basically what they use for making paper currency). It's really quite strong, and air-tight, and a few layers of it would make a decent drumhead.
Similar idea to the paper, but you'll have to add a sealer to make it air-tight. Said sealer needs to be flexible enough not to crack when it vibrates, but there are many options. Boiled linseed oil has already been mentioned. Asphalt would work (I'm not sure what there are for oil deposits in the Netherlands, but just about everywhere has some amount, even if it's not enough for modern industry to bother with) So would latex (there aren't any rubber trees in the Netherlands, but you don't need full rubber elasticity, just something close. Pine tar gets too brittle, but there's probably some plant species that would work.)
The key to a good drum is for the strike surface to transfer energy efficiently to the air. As such, you want something large and light and resilient enough not to break as it vibrates or when struck. Wood tends to get you two of the three, which is better than nothing. If you want to get all three properties, you'll need to find a tree species that is straight-grained enough that it can be carefully shaved down to almost paper thin and still have continuous fibers from one side of the drum to the other. I'm not sure what there is in the Netherlands, but trade in exotic woods has been around a very long time.
Cast glass is easy to break, but blown or spun glass aligns its structure in a way that makes it far more resilient, especially if a tempering step is added. The hard part about glass is that it's usually only resistant to being struck from particular directions, with strikes from other directions causing it to shatter. The fun part about glass is that you don't necessarily have to sound it by striking it; rubbing it with a wet hand will usually do. Imagine something big enough to have the frequency of a drum, sounded like you would a wine glass...
I know, you said, "no synthetics," and modern plastic is definitely out of the question. But humans have been making polymers of various types for a very long time. The reason leather gets hard enough to be armor when you boil it is because the natural oils in it turn into a form of plastic. The "seasoning" layer you create on a cast iron cooking pot or pan is (if you do it right anyway) a form of plastic. Linseed oil is famous for curing simply by sunlight exposure into a (very soft) plastic. All you really need is an oil (lighter is better but almost anything will do) a controllable heat source, and maybe a bit of acid. A naptha seep would do, as would a number of plant-based oils. Creating a drumhead would likely be a painstaking process of slowly layering and cooking the oil as precisely as possible. Undercook it and it'll stretch, overcook it and it'll crack or shatter, but I've made several interesting polymer compounds just playing with melting various types of tree sap over a camp fire. A culture that was dedicated to not using animal products would have an incentive to refine that process since lots of primitive waterproof containers depend at least partially on animal products and they need a substitute. The big question is whether their society is wealthy enough for anyone to have the time to make a drumhead this way. If they have no metal, then probably not unless their area is so lush they have no need to cultivate food crops or so temperate they have no need to make clothing or shelter.
This is almost the same as glass, but you're relying on finding a drumhead and maybe shaping it a little rather than creating it from scratch. Certain types of stone, especially crystals like quartz, do have reasonable resonant characteristics. The hard part is finding a piece big enough to make a decent drum. It might be something they'd have to trade for with a more volcanically active area.
Just about everything else I can think of involves and animal product or metal in some way, or is an alternate method of construction for the materials I've already mentioned.