(1) Background: Randomized controlled trials (RCTs) are considered the gold standard for evaluating a treatment. However, the results of an RCT may remain meaningless for clinical practice in cases of poor intervention feasibility or fidelity (the extent to which the protocol was executed), or when health care professionals or patients experience the intervention as irrelevant or unpleasant. Feasibility and relevance of psychosocial interventions are highly understudied in adults with autism spectrum disorder (ASD). In order to put the effects revealed in an RCT on an animal-assisted therapy (AAT) program in adults with autism spectrum disorder (ASD) into the context of clinical practice and to formulate guidelines for potential improvements and further implementation of the therapy, the aim of this process evaluation was to gain insight into the relevance and feasibility of the intervention and barriers and facilitators to its implementation. (2) Methods: Data were collected from 27 participants with ASD and three therapists using questionnaires, semi-structured interviews, and treatment reports. Reach, adherence, program fidelity, and program appraisal were evaluated, and barriers and facilitators to recruitment and implementation of the AAT program were explored. (3) Results: The participants were satisfied with the program and evaluated it as feasible and relevant for adults with ASD. The participants documented improving self-insight, joy, relaxation, and physical contact with a therapy dog as the reason of their positive appraisal of the therapy. Documented aspects that may influence feasibility and appraised relevance were the participants' therapy attitude, skills for generalization, and severity of contextual problems (e.g., problems at work, relationship problems). Regarding the sample quality, females and dog owners were slightly over-represented in the RCT. (4) Discussion: Considering the positive evaluation of the intervention and its positive effects revealed in the RCT, the AAT program can be added to the treatment repertoire to reduce stress and improve social communication in adults with ASD. More research in larger samples is needed for better understanding the generalization of the intervention effects, especially in male patients and those who do not have a dog at home.