Pretty simple, really. Create the shape you want in Illustrator. Make it a filled solid.
Import the shape into Invigorator.
Apply the bevel of your choice.
Create a new material in the material editor. Import an image of a wood surface texture into that material. You can find free textures in several locations on the web. Apply that material to your 3D object. For added realism, create a bump map
Alternately, you can create your wood grain texture as another layer in After Effects and use the layermapping technique to apply the new material and bump map to the 3D object.
Note that you'll probably need to create another wood material for the edge profile because it wraps edges differently than faces. To avoid distortion, the aspect ratio of the image being applied should be the same as the aspect ratio of the object face... or as the aspect ratio of the edge, depending on which part you're working on.
Create the other elements that go on the plaque separately.