I'm not sure humidity would be a large issue with modern papers unless you are in a very dry locale. Even so, you probably don't have to be scientifically precise - opting more for a general range of humidity. Not quite sure what your intention is, but since humidification is the issue, have you tried searching for similar concepts related to tobacco and musical instruments? In each case, people have come up with some good home-made humidification solutions you may be able to scale-up to the size you want.
Here are just some ramblings from someone (me) with no understanding of chemistry and only a cursory understanding of the other concepts involved. Take it with a grain of salt, so to speak:
Your chamber could be a larger cooler, or any container that is fairly well sealed (even a plastic cabinet).
Once your chamber is up to the correct Relative Humidity (RH), it should maintain it throughout the container - though opening the door will change things. Once the door is closed, it should reach an equilibrium again.
The cheapest way to do it would be low tech and require monitoring. You will want a decent hygrometer so you can keep an eye on the RH. After some trial and error, you can somewhat control RH by adjusting the orifice of the vessel containing your liquid. The amount of exposed surface area of the liquid can contribute to the amount of humidity; in other words, you would adjust how much of the tray of liquid was covered. This can control the rate of moisture exchange into the air. It needn't be a tray. A jug, jar, or pitcher could also work. Desiccants, scraps of cardboard, etc., can all be used as crude way of maintaining a certain RH.
I think the main thing about a humidor, aside from controlling the humidification, is using a liquid that will not easily contribute to mold/fungus. Many cigar smokers will use distilled water and change it regularly. Some use a solution of propylene glycol in an humidor. It may be possible to use polyethylene glycol, which is the main ingredient in things such as Miralax; I'm thinking cheap, do-it-yourself ideas, and have no idea if this would work.
You could also keep each pack of paper in it's own container (I know you'd prefer not to) and use something like a "
Dampit" in each container. I use these in my classical guitars. Another idea is a "
Water Pillow." Some people have created homemade humidifiers based on the Water Pillow idea using some form of absorbent kitty litter that has the same materials. For my pipe and cigarette tobacco, I came up with cheap solution using empty
Tic Tac containers with a piece of water-soaked antibacterial sponge placed inside.
There are many people more knowledgeable than I on this forum, so if they say I'm wrong, they are most likely correct