This paper introduces Whack-a-mole Online Learning (WamOL), a novel Physics-Informed Neural Network (PINNs) framework for calibrating the intraday Implied Volatility Surface (IVS) from sparse and uneven market data. The IVS calibration is formulated as a multi-objective optimization problem that requires fitting observed option prices while satisfying a partial differential equation (PDE) derived from the local volatility model and no-arbitrage inequality constraints on option price derivatives. WamOL extends standard PINNs by incorporating derivative-constrained PINNs (DCPINNs) to handle inequality constraints and introduces a self-adaptive loss balancing mechanism that dynamically reweights each loss term during training. Additionally, WamOL employs online incremental learning to update the model as new intraday data arrives, avoiding full retraining and enabling real-time adaptation. The 'whack-a-mole' strategy refers to the adaptive focusing on violated constraints, similar to the game of whack-a-mole. Experiments on real market data demonstrate that WamOL outperforms baseline methods in terms of accuracy, constraint satisfaction, and computational efficiency. The calibrated IVS yields smooth and financially meaningful risk sensitivities (Greeks) via automatic differentiation. This work provides a practical solution for real-time financial modeling, extending the application of PINNs to dynamic environments with streaming data and multiple constraints.