Numbers 6:3-4 explains that a Nazirite can't have strong drink nor vinegar of wine also can't eat anything from the vine tree or eat grapes so does that apply to any other foods with vinegar in it such as mayonnaise or ketchup?
In any case, since wine or wine vinegar are forbidden, they cannot be eaten either when mixed into other products. If one has mixed them (e.g., by mistake) and the taste of wine/vinegar cannot be felt, then it is permitted to eat the dish, but that is only valid after the fact due to the rule of "ein mevatlin isur l'chatchila".
This can be seen in the words of the Rambam (MT Nazir 5:6)
If wine or the like were mixed together with honey and the flavor of wine cannot be detected, it is permitted for a nazirite [to partake of the mixture].
From the language ("if they were mixed") one understands that this is only permitted after the fact (b'dieved) but forbidden in principle (l'chatchila).