Yes, VSP does support texture mapping.
First, select all the components in the geometry browser and change the view to 'Texture'. It will look the same as if you selected 'Shade'.
Then, go under the View menu, and pick 'Texture...' That will bring up a GUI that will allow you to place textures on VSP components. For starters, I would experiment with the texture files that come with VSP. They are in the examples/textures directory. Every airplane looks better with sharks teeth.
VSP supports *.jpg and *.tga files as textures. I'm not positive, but they may have to be power of 2 and square in size (16x16, 32x32, etc). Only *.tga supports transparency.
In addition to the texture mapping, VSP allows you to load a background *.jpg file. It could be an airport scene, or a beautiful sunset.
If these features aren't enough to achieve the results you'd like, then you should look at the PovRay export capability from VSP, or possibly some of the other file formats. PovRay is a very capable ray tracer, and there are other file formats in common with Blender and other 3D design tools.
If you get the hang of it, it would be great to see a short tutorial and some examples of working with textures added to the OpenVSP Wiki.
Good Luck,
Rob