I am making vinegar from non mevushal wine by letting it stand for several months (more than 9-12 months). After the wine has become vinegar, can a non-Jewish person move the bottle that contains it?
After the wine is vinegar it is no longer something that a gentile would pour in front of his avoda zara. Therefore, even if a non-Jew touches it, it is permitted to use. There is some controversy, at what point wine is considered vinegar for this, however if it sat so long, it is surely considered vinegar. As a side point this does not mean that we may buy vinegar from a non-Jew, because since his wine already became Yayin Nesach, and forbidden to use, even after it became vinegar, it is still not permitted. Your question is when the wine was the Jews, until after it already became vinegar, then it is permitted.
שלחן ערוך יורה דעה ס’ קכ”ג סע’ ו’ וז”ל ” חומץ שמבעבע כשמשליכין אותו על הארץ אינו נאסר במגע גוי”.וכן בס’ ק”ל סע’ ג’. וע’ רמב”ם הל’ מאכלות אסורות פי”א -י”ג וז”ל “החומץ של עכו”ם אסור בהנייה מפני שנעשה יין נסך קודם שיחמיץ”.