It depends
These sauces almost always contain conventional soy sauce, which is roughly half wheat, and many also use wheat flour as a thickener. Gluten-free versions exist and say so clearly.
A family of thick Chinese and Japanese sauces — oyster, hoisin, teriyaki, black bean, char siu — that share a structure: a savoury base, sugar, a thickener, and very often soy sauce. CiCi groups them because the gluten question has the same answer and the same cause for all of them.
Because the wheat is one layer down. These sauces are built on soy sauce, and conventional soy sauce is close to equal parts soybean and wheat. Nothing in the names oyster, hoisin or black bean suggests a grain, and the ingredient list often says soy sauce rather than wheat — so the reader has to already know that the first implies the second.
Gluten-free versions of all of these exist and are easy to find, so the home case is solved by buying once. Eating out, treat any glossy brown or dark sauce as soy-sauce-based until told otherwise — and note that teriyaki in particular is almost definitionally soy sauce plus sugar, so there is no version of it that is safe by default.
Because the answer depends on the rest of the label, scanning the actual product is the only way to resolve it — CiCi checks for a gluten-free claim, the allergen statement, and which agency regulates that label, then tells you which way it lands.
Every one of these is the same ingredient. The number of names is a large part of why gluten slips past people who read every label.
Point CiCi at the barcode on the product you're holding and get this same reasoning in about two seconds, plus the carbs and the full nutrition panel. 360,000+ packaged foods are already read and waiting.
The aisle guides below cover Oyster sauce in context — which is usually the more useful question, because an ingredient rarely appears alone.
This answer comes from CiCi's ingredient lexicon — the same rules the scanner uses, not a separate article. Where reputable sources disagree, CiCi takes the conservative position and says so.