How to stain wood veneer
You would need a stain that would penetrate the wood veneer. A good oil based stain should be able to penetrate all the way through. Lightly sand the face of the veneer with 220 grit or higher sand paper. I use a sanding block made out of rigid foam and sand by hand. You will need to keep the veneer wet with the stain until it penetrates. I would stain both sides of the veneer until it penetrates to the depth you desire and then wipe off any remaining stain. You may need to press the veneer until it dries thoroughly! Good luck.