Fog computing leverages the task offloading capabilities at the network's edge to improve efficiency and enable swift responses to application demands. However, the design of task allocation strategies in a fog computing network is still challenging because of the heterogeneity of fog nodes and uncertainties in system dynamics. We formulate the distributed task allocation problem as a social-concave game with bandit feedback and show that the game has a unique Nash equilibrium, which is implementable using no-regret learning strategies (regret with sublinear growth). We then develop two no-regret online decision-making strategies. One strategy, namely bandit gradient ascent with momentum, is an online convex optimization algorithm with bandit feedback. The other strategy, Lipschitz bandit with initialization, is an EXP3 multi-armed bandit algorithm. We establish regret bounds for both strategies and analyze their convergence characteristics. Moreover, we compare the proposed strategies with an allocation strategy named learning with linear rewards. Theoretical- and numerical analysis shows the superior performance of the proposed strategies for efficient task allocation compared to the state-of-the-art methods.