AFAIK, Zombies aren't vegetarians. They usually have a special interest in eating living human brains. Why do they eat the Plants then?
Nobody knows exactly what eldritch force or demented science allowed the dead to rise from their graves and walk. What we do know is that the resulting creature, popularly termed a "Zombie," is a restless, mindless, tireless being driven by nothing other than its hunger for brains. Your brains, to be precise.
In order to get those brains, zombies are willing to chew through any plants in their way. Though slow, zombies are persistent, and if even a single zombie gets past your defenses and into your house, he will eat your brains.
First of all, you don't have to be neither a vegetarian or a vegan to eat plants. Vegetarians are people who merely do not eat meat. Omnivores eat everything (well, a diversity of things).
In the universe of the game, the plants defend the precious brain of the person(s) in the house. The game's zombies show a basic intelligence (contrary to the classic Romero/Brooks Zombie). This leaves us with two possibilities.
The zombies don't actually eat the plants -- they just try to destroy them as they seem to block their path (this explanation would not work for the digger-zombie, though, but maybe the digger-zombies are vegetarians and want to help their carnivore friends get some brain, but that's a hell of a stretch).
They are fine eating plants as long as they are supplied but they start eating brains when the plant supply runs low. This would mean they're true omnivores.
The Zombies aren't interested in the Plants, really. They want to eat your brains. You're planting the Plants in order to prevent the Zombies from getting in your house. The Zombies say as much in their many Notes they send you between levels, and when they mutter "Brainz...." every so often.