Microservices are gaining popularity as an architectural paradigm for service-oriented applications, especially suited for highly dynamic contexts requiring loosely-coupled independent services, frequent software releases, decentralized governance and data management. Because of the high flexibility and evolvability characterizing microservice architectures (MSAs), it is difficult to estimate their reliability at design time, as it changes continuously due to the services' upgrades and/or to the way applications are used by customers. This paper presents a testing method for on-demand reliability estimation of microservice applications in their operational phase. The method allows to faithfully assess, upon request, the reliability of a MSA-based application under a scarce testing budget, at any time when it is in operation, and exploit field data about microservice usage and failing/successful demands. A new in-vivo testing algorithm is developed based on an adaptive web sampling strategy, named Microservice Adaptive Reliability Testing (MART). The method is evaluated by simulation, as well as by experimentation on an example application based on the Netflix Open Source Software MSA stack, with encouraging results in terms of estimation accuracy and, especially, efficiency.