I ask this question because wings are pretty thin overall, but I am curious if you could make them more thin.
The wing thickness is primarily dictated by structural demands. It has to provide an aerodynamic fairing for the wing spar. Making it thinner would drive up the mass of the structure inside.
To simplify things, imagine the wing spar as an I-beam. It has two flanges which carry tension and compression loads, and a web between the flanges to transmit shear loads. The bending moment carried by the spar is the product of the forces in the flanges and the distance between them. If you reduce the distance, the forces need to increase to keep the bending moment constant. Bigger forces will result in bigger flanges to keep the stresses in the material below the yield stress.
Limiting the thickness is the desire to fly at a certain Mach number. A moving wing needs to push the air ahead of it aside, and thicker wings need to do more pushing. This results in a higher acceleration of the flow around the wing, and thicker wings have a lower critical Mach number (the flight Mach number when local flow reaches Mach 1). This drag penalty is especially severe in supersonic flight.
Thicker wings also help with creating higher maximum lift coefficients, up to a point. The plot below (taken from this website) shows that an airfoil thickness of 12% gives the best results, which helps to keep the wing area low. A thicker wing also makes it easier to integrate complex high-lift devices, which again helps to reduce wing area.
The resulting wing thickness is always a compromise, and the fuel volume is only the result of that compromise. If a wing needs to store more fuel, the designers will choose a lower aspect ratio, but will leave the relative thickness unchanged. If less fuel is required, the tanks will become smaller, and again the thickness will not change.
Airplane wings are designed purely from aerodynamic and structural considerations. Supersonic fighters often have much thinner wings (and store fuel in the fuselage) because those are the requirements of supersonic flight. For subsonic (and transonic, which most modern airliners and business jets are -- over some parts of the wing the airflow is actually supersonic) flight, thick wings are fine, are structurally much better, and provide space for fuel tanks and the wheels.
Possibly, but having no fuel in the wing would likely be an effect of making wings thinner, rather than a driving cause.
In the past, thin airfoil theory ruled the day, and designers used external trusses to support two lifting surfaces in a biplane configuration. Check out the De Havilland Dragon, which used the RAF 15 airfoil section (max t/c of 6.5%), as an example. Compare this to the Boeing 247, which was introduced one month later (May vs. April 1933) and also carried ten passengers, but used a single cantilever wing, without external supports. Its wing section, the Boeing 106 (max t/c of 13.5%), was thick enough to contain an internal support structure.
The main benefit of having all your structural members contained in the wing is the reduced drag. You get some other benefits as well, like delayed stall due to the rounded leading edge, and room to store your fuel due to the higher enclosed volume. It's unlikely that any serious airliner design is going to return to a biplane configuration.
Today, thanks to the wealth of materials and design tools the engineers have, the wing thickness seems to be around 10% based on glancing at a few Airbus/Boeing offerings. Usually, a modern design will change the airfoil profile from root to tip, so it will be higher in some places and lower in others. It's possible that if these techniques were applied to biplane designs, we could come up with some really thin wings, but the airplane would have high drag and likely wouldn't sell.
That said, some concepts arrange the wings in ways that might let you get away with this. In particular, Lockheed Martin has proposed a box-wing concept (for which I couldn't find the name) and Boeing has studied a design with braced wings called Subsonic Ultra Green Aircraft Research (SUGAR). If one of these designs ends up reducing the amount of stiffness needed in the wing spar, the designers would likely reduce the spar (and thus wing) thickness to save weight. Depending on how much thickness they shave off, it might no longer be worthwhile to store fuel in the wing.