The growing deployment of Internet of Things (IoT) technologies has enabled highly distributed cyber-physical systems in which the sensors and controllers are physically separated and operated by distinct entities who may not able to coordinate. In this work, we formulate a game-theoretic design framework to capture the non-cooperative behaviors between a sampler and a controller. At the cyber layer, the sampler aims to find the best sampling scheme to achieve its design objective. At the physical layer, the controller aims to solve a class of linear-quadratic Gaussian (LQG) problems subject to the arrivals of the sampled observations. The system performance under the uncoordinated sampling and control can be characterized by the Nash equilibrium of the nonzero-sum game. We completely solve the controller's problem under a given information structure and provide a sufficient condition for the game to admit a unique equilibrium. Under mild conditions, we show that the sampling scheme at the Nash equilibrium results in a worse performance of both the sampler and the controller due to the lack of coordination. We use numerical examples to corroborate the analytical results and show that there exists a fundamental threshold on the sampling rate for an unstable system to be stabilized.