Can vampires turn people into zombies?

Probably not, cuz when a vampire is alive it doesn't mean that you he can control you or something.... They might turn to you into vampires with a bite or in another way. The reason why is that, because they are not Zombies them selves. So they might only have special vampire features and actions but it can't turn you into a zombie...

As an ordinary person (with the right knowledge) can create a zombie there is therefore no reason why a vampire (with the same knowledge) could not do the same.

The process to become a vampire is a delicate one and if it is not conducted properly then the results may be 'less than perfect'. It is therefore possible that a poorly created vampire might be indistinguishable from a zombie: incapable of stealing the vital life-force that it endeavors to obtain through consumption of prey, decomposing, very low intellect.