You need a source of energy. My first thought was an elastic band, twisted many times, so that on release it would untwist and power your wheels, but you seem to rule that out.
Internal combustion engines use the expansion of a gas, so a chemical reaction that causes a material to expand would fall in the allowed category. You do not even need a chemical reaction: think compressed air. If you can get a small CO2 tank like those used in Paint Ball, you can make it release gas to a thread wheel.
Speaking of which, you can put mass, solid or liquid, at some height and use its downward movement to propel the egg support a small distance, This may be hard because the mass and height cannot be large, and may not meet the energy requirements for a 8 meter dash.
I'd stick with the CO2 container. Hope this helps. Tell me if it does.