The purpose of oyster sauce is to add savory and salty flavors to your food. A proper vegetarian oyster sauce will do this too, often with the help of mushrooms. There are many different brands of vegetarian oyster sauce, and they don't always taste the same, so you may end up having to adjust the seasoning of your dish.
Look for vegetarian oyster sauce at an Asian market. If you aren't able to get ahold of any, you can just use soy sauce as a substitute.