It depends
Commonly soy or corn, but wheat is used too. If it were wheat on a US food label it would have to say so — which makes this lower risk in the US than it is often given credit for. The residual concern is barley.
Plant protein broken into fragments with acid or enzymes so it tastes intensely savoury. It is a flavour booster, chemically close to what fermentation does to soy over months, done in hours. "Vegetable" is doing no work in the name — it only means not animal, and the plant is unnamed.
Because the plant used to be wheat as often as anything else. Historically HVP in the US was frequently wheat-based, and people who learned the rules a decade or two ago learned to avoid it. Soy and corn have since become the usual sources, but the ingredient name still does not say, which is why the old caution persists and why it is not unreasonable.
On an FDA label with no wheat declared, fine. On a USDA product, the word alone does not settle it and a gluten-free claim does. Note that the related term "textured vegetable protein" usually means soy and is normally fine, while "textured wheat protein" is what it says it is.
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.
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.