Synthetic autonomous locomotion shows great promise in many research fields, including biomedicine and environmental science, because it can allow targeted drug/cargo delivery and the circumvention of kinetic and thermodynamic limitations. Creating such self-moving objects often requires advanced production techniques as exemplified by catalytic, gas-forming microrockets. Here, we grow such structures via the self-organization of precipitate tubes in chemical gardens by simply injecting metal salts into silicate solutions. This method generates hollow, cylindrical objects rich in catalytic manganese oxide that also feature a partially insulating outer layer of inert silica. In dilute H2O2 solution, these structures undergo self-propulsion by ejecting streams of oxygen bubbles. Each emission event pushes the tube forward by 1-2 tube radii. The ejection frequency depends linearly on the peroxide concentration as quantified by acoustic measurements of bursting bubbles. We expect our facile method and key results to be applicable to a diverse range of materials and reactions.