I'm going to make a growth height chart for my daughter just like this one.
https://www.amazon.com/Back40Life-Wooden-Growth-Chart-Establishment/dp/B08DDFTZ77?th=1

I already have the stencils and my dad has a nice piece of dark walnut I can use. I'm not sure how to finish the walnut or if I should just leave it natural. Obviously it will be permanently inside. If I lacquer the piece after the stenciling is done, I'm afraid the white paint will turn yellow over time. If I lacquer it first, will the stenciling stick on top of it? I thought about just applying a finishing wax over it, which I think will look nice... but will I still be able to write on it with a Sharpie every year or will the wax not let anything stick to the surface?

Try linseed oil or Watco Danish oil finish

You can use all kinds of finish on walnut. Do a test piece with the different finishes and sheens and see which one you prefer. I prefer a satin finish on walnut

So I can apply the Danish oil after the stenciling and once it's dry, I'll be able to write on the wood?

I would paint the stencil it first, then go over everything with a couple coats of poly. Whatever gloss level you like. Easy peasy.

Once the poly cures you can write on it with a sharpie no problem.

Any clear finish can yellow a little given enough time, but poly is going to be much more stable than most. And most "natural" oils like linseed, danish, and tung oils never really fully cure 100% so I would be worried about your future sharpie marks bleeding into the wood grain.

Clear/natural Oil finishes are actually amber colored. There are water based finishes that are white when you put them on but they turn clear as they dry.