Speaking as a former UPS truck loader, I wholeheartedly recommend Brown for your vape-shipping needs. As long as you have it well-packaged (we didn't try to damage parcels, but when you're moving several thousand pounds per shift, sometimes damage just happens), and you fill out the shipping label legibly, UPS will get it there with no questions asked. I'd label it "toys" or "portable electronics" or something equally innocuous if you're really concerned about anyone giving your package a second look. Honestly, though, I don't think anybody in the shipping chain--from the receiving clerk to the delivery driver--actually cares at all about what you're sending.
USPS is probably the same, but I don't have first-have experience with them so I can't say for certain.