Create an installer that ensures this dependency is met. WiX is the normal route here. The WiX tools do not depend on VS, although the VS tools make things a little easier for you.
-or-
In VS, open the references section of your project, find the reference to MySql.Data, go to properties and set "Copy Local" to true
and rebuild. Now the .dll will be copied to your bin
folder, ready for redistribution as you see fit.