Primary symptoms of adults with autism spectrum disorder (ASD), such as pervasive social deficits in social interaction and communication, cause adults with ASD to adopt a sedentary lifestyle. Meanwhile, gamified and behavioral theory-based interventions have been shown to improve physical activity in a fun and unobtrusive way. In this paper, we describe the iterative design inquiry process of PuzzleWalk, a gamified, physical activity-promoting mobile app designed for adults with ASD. We report the design rationales and lessons learned across four user-centered design phases with ASD experts and adults with ASD, including user requirement gathering, iterative participatory design, usability evaluation, and field deployment. The design insights generated from this work could inform future research focusing on designing sociotechnical systems, games, and interventions for people with ASD.