The Vichy French government had an interesting dilemma, while many Vichy leaders didn't have the outright contempt for the Jews as the Nazis did, they also had to walk a fine line between collaboration and defiance, less their fragile independence be ended. Vichy France, despite having de-jure control over the entirety of France, only had de-facto control over their little bit of territory in the South of France. The Germans exercised complete control over the North and thus Vichy France could do nothing to stop the Germans from persecuting the Jews in the North.
Now in the South, where Vichy France still held power, the Vichy French initially tried to avoid any sort of collaboration, but a number of pro-collaboration proponents gained power. Chief among them Pierre Laval and Joseph Darnand (who commanded the militia that would turn into the Milice).They did enact measures against undesirables, specifically Jews, métèques (immigrants), Freemasons, and Communists, there were a number of proponents for these measures, as they assumed it would help make France strong again. These groups also received blame for France losing the Second World War. This is when the Jews began to be serious repressed, and measures introduced in Nazi Germany (the yellow star, segregation, etc.) began to be introduced in Vichy France starting in late 1940. Pierre Laval complained about it in secret to the Italian ambassador, but he did little more than complain.
These measures never got as serious as the ones in Nazi Germany, but they were still brutal. In 1939, Drancy Interment camp was founded and was used to hold undesirables. While Vichy France would remain in control of this camp until 1943 (when it was taken over by the SS). In 1942 the Germans began to pressure their various satellites (Slovakia, Hungary, etc..) to begin large scale deportation of Jews. The French eventually succumbed to the pressure, and the first trains, filled with Jews, left France on March 28th 1942.
Any semblance of independence for Vichy France came to an end in November of 1942. The Germans and Italians enacted “Case Anton”, which was their plan to occupy Vichy France to protect their Southern Flank from allied invasion. The Germans moved into French territory and thus were able to begin full scale persecution of the Jews. Hitler issued this order:
For all tasks which concern the Reichsführer SS and Chief of the German police…. In these areas of competence he has the right to order and inspect French agencies and police units. He may deploy the French police forces in the occupied territories
Many Jews fled to the Italian occupation zone (with the consent of the Italian government), where they found relative safety until the fall of Italy in 1943. Pierre Laval put up token resistance to the harsher treatment of the Jews, but with Vichy France occupied by German troops there was nothing he could o except complain. He argued that the treatment of the Jews was a violation of French sovereignty and that any actions taken by the Germans should be double checked with the “Marshal” Phillipe Petain, still the de-jure head of Vichy France.
So overall while the Vichy French tried to put up some token resistance in areas, it was more to assert their independence than out of actual compassion for the Jews. Some sources on this are “All or Nothing” by Johnathan Steinberg. I also found this article interesting