In this work, vector framework based contraction theory to design interval observers for a class of nonlinear systems having disturbances, inputs and outputs is exploited. The main feature is that it does not require the formulation of error dynamics to show the convergence properties and need not require the construction of a Lyapunov candidate function without any idea of the structure of the function. Specifically, it performs the convergence analysis through a comparison system which has specified properties. Furthermore, this theory is exploited to design dynamic output feedback control through the obtained state bounds from the constructed interval observer and the system outputs, to make the interval observer to be globally asymptotically stable. Examples with simulation outcomes are provided to validate the theoretical results.