(I edited this answer; instead of clay batons, now we have a "clay sheet". The reason is that sheets have a bigger surface and actually would be more practical than several batons)
I think they could do it.
First of all, I'm not an expert in biochemistry or anything like that. I just made some research, got some names and reactions, and assumed we have some creative license here.
About the reaction
The bioluminescence in fireflies is created by the reaction of:
- luciferin, a chemical present in the fireflies;
- oxigen, that will react with the luciferin;
- luciferase, an enzyme which acts as catalyst (and thus is not consumed by the reaction).
Basically, to create light, you need to create a reaction between the oxigen and the luciferin. That won't happen by itself, and that's why luciferase is on the list: it will catalyze the reaction and make the reagents release energy, making stuff glow.
Getting your ingredients
(Again, I'm nowhere close to be an expert in (bio)chemistry, so I'll use some artistic license we have in fantasy/science fiction)
The source of the oxygen will be the air itself.
The source of luciferin and luciferase are the fireflies. You can invent the process you want to get them, but I'll guess mashing the insects with the right materials will make the thing and conveniently "trap" the chemicals you want.
An example of process could be:
- Getting the luminescent organ from a firefly;
- Envelop it in a convenient kind of clay that will trap the luciferin;
- Put them inside a recipient filled with a convenient oil or liquid (so there will be no air to generate the reaction);
- Mash your clay + luminescent organ until you're sure you have just a little luciferase in your mash;
- Remove the mash from the oil. It may glow a little, but since you are a very dedicated person, the reaction will be brief;
- Continue mashing the clay out of the oil until it stops glowing;
In the end, we'll have an oil containing luciferase and luciferin trapped in the clay. Of course, this is only for the convenience of crafting the lamp, what comes now.
To make the lamp you want, you'll make a "sheet" of clay (not sure if it's the name of the object, but it's a rectangular flat piece of clay), that you'll put inside the lamp. Maybe it needs to dry to get firm.
The lamp can be made of wood, metal, or even glass. It would be similar to a common lamp you carry around, with holders to put the sheet in. It also has two little recipients: one on the top of the lamp and one in the bottom. If you opt for a "closed" design, it needs to let in some air.
The recipient on top will be filled with the oil you mashed the clay in; it will need some kind of mechanism to control the oil flow. The oil, as you may guess, contains luciferase and will flow on the clay sheet, creating the reaction. The flow mechanism lets you select how much of the sheet you want to light.
The recipient at the bottom will be used to capture the oil that will flow in your lamp, since luciferase is not consumed by the reaction and could be reused.
To make the lamp work, you'll have to:
- Put a new sheet of clay inside the lamp;
- Fill the upper recipient with oil;
- Open the mechanism and let the oil flow along the sheet; the oil will make the luciferin in the clay react with the oxygen in the air, making it glow (open more to light a bigger surface of the sheet and get a brighter light).
- Change the clay sheet when the reaction ends(luciferin will be consumed, so this sheets are kinda like batteries);
- Collect the used oil to reuse it.
And basically that's it. I've made some sketches, I hope it helps you to visualize the idea.