Statistical process control techniques have been widely used for online process monitoring and diagnosis of streaming data in various applications, including manufacturing, healthcare, and environmental engineering. In some applications, the sensing system that collects online data can only provide partial information from the process due to resource constraints. In such cases, an adaptive sampling strategy is needed to decide where to collect data while maximizing the change detection capability. This paper proposes an adaptive sampling strategy for online monitoring and diagnosis with partially observed data. The proposed methodology integrates two novel ideas: (i) the recursive projection of the high-dimensional streaming data onto a low-dimensional subspace to capture the spatio-temporal structure of the data while performing missing data imputation; and (ii) the development of an adaptive sampling scheme, balancing exploration and exploitation, to decide where to collect data at each acquisition time. Through simulations and two case studies, the proposed framework's performance is evaluated and compared with benchmark methods.