I'd have to say no. Here's my thinking:

Vampires are corpses that drink blood, regardless of how they are created - from the bite of another vampire, from drinking the blood of a vampire, from a ritual or a curse. The most common depiction is that they are dead, but continue to live and stay young and have magical powers, and drink blood to maintain that state.

Zombies can happen in a few ways:

1) Virus - some kind of disease makes people insane and wanting only to bite to pass on the disease. In some cases the disease (logic aside) keeps the corpse moving as well. Vampires are corpses, their bodies cold, their organs atrophied, their hearts still. All but the most dramatic of wounds, or those cause by sunlight, fire or, sometimes, holy items, seems to close instantly on them. In these conditions it's safe to say that no disease would be capable of being passed to them. The body is dead and disease need living bodies to thrive in most cases, the body instantly heals all outside interference, which I have to assume includes disease, and if vampires were capable of having and spreading diseases then their blood drinking would be almost instantly exposed because they would be constantly passing on blood borne diseases every time they fed.

2) Magic - a lot of zombies appear to be corpses capable of movement and fighting. Bodies damaged beyond the point of being mobile continue to fight and kill. Rotten muscle tissue, missing organs, exposed bones... None of it seems to stop the zombie from moving. This is magic. You can't move your hand if the rest of your arm is only bones, all the required tendons, muscles, blood etc are simply missing. The only answer is that some magical necromancy is moving you. In this case, I'd have to say that whatever magic is creating the vampire wouldn't be able to also make it a zombie. If you're a zombie and someone tries to raise you as a zombie, you don't become two zombies, right? Why would a vampire, a creatures that's already raised from the dead, rise from the dead again? Can skeletons be ghosts? Can mummies be wights? I'm going to assume that you can only be one kind of undead at once.

3) Radiation, aliens, experiments, etc - There's a handful of other, less common forms of zombies in media. Sometimes a meteor brings radiation that raises the dead. Sometimes a scientist has a formula that does it. So on, so forth. In almost all these cases the requirement is that the person be dead in the traditional sense. Sometimes it's so picky it won't even work on the corpses of animals, only people. I have to assume that, in these rare situations, the vampire would be immune due to it's magical, pre-undead nature. If your space rock radiation isn't raising dead trees and dead rats, then it seems far to specific and specialised. How could it hope to compensate for something as complex as an already dead person who remains animated through magic and blood drinking?

Lastly, I think it's just a question of storytelling. Most stories are served better by keeping their types of undead distinct. Vampires are vampires, zombies are zombies. They serve different narrative purposes and fit different story beats and stereotypes. If your story is specifically about zombie vampires, then go your hardest... But what does that really serve? Do the normally intelligent and cunning alpha-predator vampires become slow, shambling, unintelligent zombies? If so, can they still use magic, like turning into bats or hypnotising people? Why can they do that, if they are now brain dead zombies? Doesn't that mean that a zombie vampire is, in essence, just a zombie? Why bother doing that, then?

Maybe if you're writing a book about vampires having to combat a zombie plague because they aren't immune? Even then, it might work as a comedy, but you're really asking the reader/watcher/player to accept a lot by heaping zombies, vampires and zombie vampires that force vampires to work with humans against zombies all into one story. It'd be a lot of hard work to make a compelling, believable, interesting and serious work out of it... But not so hard to make it a kind of goofy lark.

I should also say that it takes 0 creativity and effort to ignore all of the above. You can simply says: "Some diseases live in dead bodies, so this one does"... You can always find or create an exception to ignore the rule. Heck, even just: "Well in my story it does" works. These were just my thoughts as to why, as a logical rule, the answer would be no. :)