Magnonic excitations in the two-dimensional (2D) van der Waals (vdW) ferromagnet chromium triiodide (CrI3) are studied. We find that bulk magnons exhibit a nontrivial topological band structure without the need for Dzyaloshinskii-Moriya interaction. This is shown in vdW heterostructures, consisting of single-layer CrI3 on different 2D materials such as MoTe2, HfS2, and WSe2. We find numerically that the proposed substrates substantially modify the out-of-plane magnetic anisotropy on each sublattice of the CrI3 subsystem. The induced staggered anisotropy, combined with a proper band inversion, leads to the opening of a topological gap of the magnon spectrum. Since the gap is opened nonsymmetrically at the K+ and K-points of the Brillouin zone, an imbalance in the magnon population between these two valleys can be created under a driving force. This phenomenon has a close analogy to the so-called valley Hall effect and is thus termed the magnon valley Hall effect. In linear response to a temperature gradient, we quantify this effect by the evaluation of the temperature dependence of the magnon thermal Hall effect. These findings open a different avenue by adding the valley degrees of freedom besides the spin in the study of magnons.