I’m looking for shoulder screws that have very little slop and are around 1.5 to 2 inches long, with the “smooth section” at or over 1 inch long. Does anyone know where and how to look for these dimensions? I’m not very familiar with screw sizing 
Off topic, but will shoulder screws eat away joints like axles do when they’re installed without bearings?
Go to McMaster Carr.
Would this work with existing nuts and screwdrivers/be competition legal?
Type 316 Stainless Steel Shoulder Screw, 3/16" Diameter x 1-1/4" Long Shoulder, 8-32 Thread
I’m also wondering if this will cause too much friction in the vex square holes as it is only slightly smaller than standard holes.
I have not been able to find shoulder screws at a reasonable price though. I am a big fan of them for holding power.
Vex: $10 per 25 but just one size. Sometimes you need another layer of metal to hold things tight. Maybe a smaller shoulder length option would be nice as well as longer shoulders.
http://www.vexrobotics.com/276-1408.html
Other places are priced much higher for shoulder screws: Check for yourself.
http://www.boltdepot.com/Shoulder_bolts_Allen_Stainless_steel_18-8_3_16_shoulder_(8-32_thread).aspx
Shoulder screws work very well in holding things together tightly but you will have a bit of friction on the metal if you want to use it as a joint. It touches 4 sides of the square hole. Good for evening out big stress points.](McMaster-Carr)
Yes, but not as rapidly as the square axles in our experience.