We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable kernels of the form
K(x,x′)=k(x,x′)T, where
k is a scalar-valued kernel and
T is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical
K=kI paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.