In this paper we propose a general framework to perform statistical online inference in a class of constant step size stochastic approximation (SA) problems, including the well-known stochastic gradient descent (SGD) and Q-learning. Regarding a constant step size SA procedure as a time-homogeneous Markov chain, we establish a functional central limit theorem (FCLT) for it under weaker conditions, and then construct confidence intervals for parameters via random scaling. To leverage the FCLT results in the Markov chain setting, an alternative condition that is more applicable for SA problems is established. We conduct experiments to perform inference with both random scaling and other traditional inference methods, and finds that the former has a more accurate and robust performance.