The most widely-accepted substitute seems to be various sauces made from mushrooms. I don't want to say "mushroom sauce" because most products don't have that exact name. For example:
Here you have "Vegetarian Mushroom Flavored Stir-Fry Sauce" on the left, and "Premium Shitake Mushroom Flavored Soy Sauce" on the right. I'm actually not familiar with the one in the middle and not sure what the whole label says - it might be the same as the first.
If you go to a specialty Asian food store, you might actually find a product called "vegetarian oyster sauce", which is pretty much the same thing.
Failing that, you can always substitute soy sauce, or hoisin sauce if you want a less watery texture. They won't be exactly the same, but it's the same general idea (dark, fermented, salty sauce rich in glutamates).
"Fish sauce" will be very close to oyster sauce in taste, but may still contain shellfish, so be careful.