Neglecting the social aspect of complex sociotechnical systems can prevent system success. For example, nuclear power plants might have excellent stability and minimal waste, but will not be easily adopted if the public opposes them. Public opinion of an engineered system can have a strong impact on its success, thus motivating the need to understand how a system is perceived. Traditionally, public opinion about technologies, such as vaccines, nuclear power, nanotechnology, firearms, etc., have been assessed using surveys. Traditional survey methods remain the gold standard for measurement of public opinion, yet they face increasing challenges. Furthermore, advances in social media and machine learning techniques have led to new opportunities to assess public opinion about complex socio-technical systems. Thus, it is important to understand how practices to gather and analyze these perceptions might be evolving. Indeed, state-of-the-art methods using artificial intelligence and social media to access opinion offer opportunities that may mitigate the challenges faced by survey techniques, yet they present their own set of concerns. We study this changing landscape in the context of one of the most widespread technologies of modern times: vaccination. Vaccines present a poignant example of how public opinion can affect sociotechnical systems: despite increased vaccine technical performance, vaccine hesitancy and refusal are becoming a global problem. This motivates the key contribution of this paper: we present a review of key concepts in survey research, relating them to prior work on the study of public opinion using social media and machine learning techniques. We compare and contrast the techniques and biases favored by each branch of literature in the context of vaccine hesitancy and other complex socio-technical systems. Finally, we conclude with a discussion of opportunities and directions for future research on how these two literatures may be synthesized, identifying gaps and opportunities to improve the measurement of public perception of vaccines and other socio-technical systems.