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Add a material if there is not one already and/or if the "use nodes"checkbox is missing.
You should have the Diffuse BSDF and Material Output connected together from BSDF to Surface. Add an input node for attribute: Add (or Shift-A) --> Input --> Attribute.
Add the name of your vertex paint color as the attribute. The default name is Col.
If you are uncertain of the name of your vertex paint color, go to the vertex tab from the properties panel. Under "Vertex color", you should be able to find the name.
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Now your vertex paint should display in your render!