Not all Jews support Zionism. Several ultra-Orthodox groups perceive Zionism as an anti-messianic act and have denounced it. However, this does not make them anti-Semites.
Many proponents that equate anti-Zionism with anti-Semitism frequently assume all Jews are allies of the state of Israel. However, many Jews do not support Zionism and are explicitly anti-Zionist in their political beliefs. But they, as Jews, cannot be considered anti-Semitic as well.  In fact, Jacob Blaustein, the president of the American Jewish Committee, reached an agreement with David Ben-Gurion, the former prime minister of Israel, in which Ben-Gurion promised not to claim he spoke for American Jews. Prominent Jewish groups such as Neturei Karta also oppose the State of Israel. They believe that the Jewish people are not to establish the Land of Israel by the use of physical force in the face of political opposition. They believe that the State of Israel will be established through divine redemption alone. Any attempt to use political means or physical force to establish the Israeli state is explicitly anti-messianic and represents a denial of divine providence.  None of these groups could be accused of being anti-Semitic. Therefore, anti-Semitism and anti-Zionism are clearly very different concepts.
It is absolutely possible for Jews to be anti-Semitic. In fact, whole university courses have been devoted to the subject of Jewish self-hatred. Throughout history there have been examples of Jews internalising the negative image others have of them and becoming anti-Semitic themselves. This is precisely what occurs when Jews subscribe to anti-Zionist beliefs. They are internalising the criticism of Israel and the attacks on Jewish right to self-determination, causing an identity crisis and manifesting itself as feelings of anti-Semitism.
[P1] Some Jews are anti-Zionists. [P2] Jews cannot be anti-Semitic. [P3] Therefore, anti-Zionism and anti-Semitism are distinct concepts.
Rejecting the premises
[Rejecting P2] Jews can be anti-Semitic.