Active, point-like, self-propelling particles -microorganisms, autophoretic colloids, and active droplets -are known to self-organise into collective states via inter-agent interactions. Here, we create conformationally free extended active objects -polymers -and show that self-shaping interactions within the polymer result in novel dissipative structures. Using experiments, simulations and theory, we show that chemical interactions between the jointed active droplets result in emergent rigidity, stereotypical shapes and ballistic propulsion of the active polymers. These traits, quantified by the curvatures and speeds of the active polymers, vary systematically with the number of monomers comprising the chain. Using simulations of a minimal model, we establish that the emergent properties are a generic consequence of quasi two-dimensional confinement and chemical interactions between the freely jointed active droplets. Our work provides insights into the emergent organization of extended active objects due to self-shaping interaction landscapes.