This paper considers the problem of online trajectory design under time-varying environments. We formulate the general trajectory optimization problem within the framework of time-varying constrained convex optimization and propose a novel version of online gradient ascent algorithm for such problems. Moreover, the gradient feedback is noisy and allows to use the proposed algorithm for a range of practical applications where it is difficult to acquire the true gradient. Since we are interested in constrained online convex optimization, we carefully select the step size at each iteration so that the iterates stay feasible. In contrast to the most available literature, we present the offline sublinear regret of the proposed algorithm up to the path length variations of the optimal offline solution, the cumulative gradient, and the error in the gradient variations. Furthermore, we establish a lower bound on the offline dynamic regret, which defines the optimality of any trajectory. To show the efficacy of the proposed algorithm, we consider two practical problems of interest. First, we consider a device to device (D2D) communications setting, and the goal is to design a user trajectory while maximizing its connectivity to the internet. This problem is of vital interest, due to the surge in data-intensive applications in smartphones, and the consistent internet connectivity is becoming essential. For this problem, we consider a pair of pedestrians connected through a D2D link for data exchange applications such as file transfer, video calling, and online gaming, etc. The second problem is associated with the online planning of energy-efficient trajectories for unmanned surface vehicles (USV) under strong disturbances in ocean environments with both static and dynamic goal locations. We consider an unmanned surface vehicle (USV) operating in an ocean environment to traverse from start to destination. Different from the state of the art trajectory planning algorithms that entail planning and re-planning the full trajectory using the forecast data at each time instant, the proposed algorithm is entirely online and relies mostly on the current ocean velocity measurements at the vehicle locations. The detailed simulation results demonstrate the significance of the proposed algorithm on synthetic and real data sets.