I voted for “France could have fought with the Axis but this would not have changed the outcome of WWII”.
Vichy France actually did “fight for the Axis” on a small scale, in such forms as the 33rd Waffen Grenadier Division of the SS Charlemagne, whose last combat action was – ironically enough – defending the Führerbunker against the Russians during the Battle of Berlin. Vichy also contributed appreciably to Nazi Germany’s home front by deportee labour through such “programs” as the STO, the Service de Travail Obligatoire (compulsory work service).
Could Vichy have fought on the Axis side on a larger scale, meaning as a full belligerent power? I think the short answer would be “possibly, but not probably.” I don’t think France would have joined the Axis in this manner as the result of any German inducements; I think that, if it was going to happen at all, it would have been as a result of France finding its own reasons to go to war against Britain (the only Allied power still in the game in the second half of 1940). The only casus belli which might have reached that threshold was the British attacks against the French navy at Mers el Kebir and Oran – attacks which qualified as acts of war. France ultimately chose not to go to war with Britain despite this provocation, which explains the “but not probably” part of my response. As Wittmann has said, France was in no mood for war after its 1940 defeat; indeed, it was in no mood for war prior to its defeat, which is one of the reasons it was defeated in the first place.
It should also be noted that Petain, for all of his collaborationism, saw himself as a French patriot and was more of a naive dupe than the much more cynical and opportunistic Pierre Laval (who was executed after the war, whereas Petain’s death sentence was commuted to life imprisonment). Petain wanted to maintain a fiction of French dignity and autonomy despite the surrender, an objective he made clear right from the time of his radio address to the nation when he said that he was going to contact the Germans and discuss “soldier to soldier” the terms of an armistice. He achieved this fiction through such elements of the Vichy regime as the “unoccupied zone” and the retention of the French fleet under Vichy control rather than turning it over to Germany. I don’t know to what extent he managed to fool anyone other than himself with these pretenses. After Germany occupied the rest of France following the Allied landings in North Africa, Petain spent the rest of the war pouting like a three-year-old – basically, sitting in his office with his arms crossed and refusing to govern, which was his conception of taking a principled stand. The Vichy regime as it existed prior to that time (authoritarian, agrarian, Catholic, and equipped with a personality cult that revolved around him) suited him just fine, so he had nothing to gain by jumping into the war outright as a full Axis partner.
Could full French participation in the war as an Axis partner have changed its eventual outcome? Probably not, though it would have complicated matters for the Allies. The French fleet might not have had much impact in the Atlantic, but in the more confined waters of the Mediterranean (where the Italian fleet was also operating) it might have made life more difficult for the British it it had been used aggressively (something which can be said about the Italian fleet too). On the Eastern Front, the French army would probably have been relegated to supporting roles, in the same way that the Italians, the Hungarians, the Romanians and other Axis minor forces were used. The contributions of all of those forces were, in brief, helpful but not decisive. You could even say that they were “decisive, but in the wrong way” because, at the Battle of Stalingrad, the Axis minor forces holding the front to the north and south of the German 6th Army were the weak points that the Russians attacked when they launched the two-pronged offensive which ultimately trapped Paulus.
Another point to keep in mind is that the French forces which could hypothetically have been deployed on the Eastern front in such a scenario would inevitably have been weaker and less motivated than the French army was in May 1940. They would have been weaker because of the losses they suffered in the Battle of France, and they would have been less motivated because of the humiliating defeat they had suffered, because they would not have been fighting to defend their own homeland (a job they didn’t even handle well when they were doing it) and because some of their talented and motivated officers and men had gone over to the Free French side. One also has to wonder: if the French army hadn’t been able to stand up to the highly-motivated Wehrmacht on temperate French soil, how well could it have stood up to the (arguably even more motivated) Russians in temperatures of fifty below zero?