What kind of dough staple is easiest to make, for soaking in mussel juices? No oven, eggs, milk

The easiest dough staple to make stovetop is some kind of flatbread.

Bread dough can be pretty simple, flour and salt and water and yeast, so you don't need to worry about milk or eggs - though you can try adding additions if you want to modify the results, fats to make it softer, flavoring agents, etc.

I think you'd be looking for something rolled thick, to give as much contrast as possible between a crisp crust and a soft interior. It might be helpful to cover the flatbread with a lid, to help it cook more evenly (with steam), and also keep it a bit softer.

I think you'd end up with something like naan, which has both soft and crisp areas and is quite good at mopping up sauces... though the flatbread will likely be a bit less tender and more lean since naan recipes I've seen use yogurt while you're avoiding milk.

In the southern US, people make what is known as flour bread. (Also known as biscuit bread or skillet bread.) It's made with biscuit dough that is flattened into a circle about 1/2" thick. It is cooked on top of the stove, in a lightly greased skillet over medium-high heat on one side until that side is golden brown and then flipped and cooked until golden on the other side.

I typically use leftover biscuit dough that has been shaped into a ball, wrapped and refrigerated overnight. (It doesn't have to be leftover but I do find that the time in the fridge makes a difference.)

Typical southern-style biscuits are made with flour, salt, hard fat (vegetable shortening or lard), baking powder, and water or milk (cook's choice). So, no need for milk, eggs, or an oven.

It's great for sopping up gravy or juices.

Flatbread like naan definitely seems like the easiest way to go. For the best results I recommend a dutch oven or a cast iron pan with a cast iron lid. Here are instructions for using a dutch oven with a bread pan.

Some flatbreads are cooked directly over a gas flame (at least on one side. When a colleague made something similar they were rolled out rather thicker than naan and had a nice combination of thin crispy crust and soft interior. The bread is held with tongs. This dough is made from only flour, water, and oil

Another option that might work (I've only ever done it over a campfire) is bread on a stick. This is a leavened bread with yeast, and gives you something like a hollow bread roll. Some experimenting might be needed. Indoors I'd probably use a long-handled wooden spoon unless I could find thick/long enough bamboo skewers.

I recommend good ventilation, ideally an extractor hood.

The recommendation for crusty, fluffy bread is traditional - in French restaurants. As this isn't a French restaurant feel free to choose a more suitable bread.