The ability to treat complex medical issues often requires dynamic and versatile materials. Electrospinning is a fabrication technique which produces nano-/microfibers that can mimic the extracellular matrix of many biological tissues while shape memory polymers allow for geometric changes in devices upon implantation. Here, we present the fabrication of electrospun polyurethane which exhibits the shape memory effect. To improve the mechanical and shape memory properties of this system, we incorporate vinyl side chains in the polymer backbone which enable crosslinking via thiol-ene click chemistry post fabrication. We also discuss a novel technique to improve photoinitiated crosslinking for electrospun materials. A material with these properties is potentially beneficial for various medical applications, such as vascular anastomosis, and the characterization of this material will be valuable in directing those applications.