Poor sleep health is an increasingly concerning public healthcare crisis, especially when coupled with a dwindling number of health professionals qualified to combat it. However, there is a growing body of scientific literature on the use of digital technologies in supporting and sustaining individuals' healthy sleep habits. Social robots are a relatively recent technology that has been used to facilitate health care interventions and may have potential in improving sleep health outcomes, as well. Social robots' unique characteristics -- such as anthropomorphic physical embodiment or effective communication methods -- help to engage users and motivate them to comply with specific interventions, thus improving the interventions' outcomes. This scoping review aims to evaluate current scientific evidence for employing social robots in sleep health interventions, identify critical research gaps, and suggest future directions for developing and using social robots to improve people's sleep health. Our analysis of the reviewed studies found them limited due to a singular focus on the older adult population, use of small sample sizes, limited intervention durations, and other compounding factors. Nevertheless, the reviewed studies reported several positive outcomes, highlighting the potential social robots hold in this field. Although our review found limited clinical evidence for the efficacy of social robots as purveyors of sleep health interventions, it did elucidate the potential for a successful future in this domain if current limitations are addressed and more research is conducted.