This is because Vertex Oven can only edit the original vertices of the mesh, before any modifiers. If there are any modifiers that change the shape of the mesh, Vertex Oven will bake ambient occlusion of the new shape to the underlying shape, which will be self-occluded in the wrong places.
To avoid this issue, viewport disable any modifiers that change the mesh shape (click the icon that looks like a TV, the second from the left.)
I'm developing workarounds for this, so stay tuned.
This is because Vertex Oven operates per-vertex, and cannot change the ambient occlusion on faces. The only solution to this is to add more vertices in the affected areas.
Select the mesh, open the Object Data panel (the small green triangle tab in the Properties panel), select the vertex color slot (Ambient Occlusion by default,) and click the "minus" button to the right.
No problem! If you don't want to purchase it on Blender Market, you can download it for free from GitHub.com/ForestKatsch/VertexOven. If you use it a bit and find that Vertex Oven improves your workflow, please consider purchasing it as well. Thank you for considering Vertex Oven!
(All previous versions of Vertex Oven are also available on GitHub, in case you want to stay on an old version.)