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The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome this limitation by randomized sketching combined with an integral representation of f(A)b. Two different approximation methods are introduced, one based on sketched FOM and another based on sketched GMRES. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed-form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the presented sketching approaches. [ABSTRACT FROM AUTHOR]