Martinique, Guadeloupe, and French Guiana were all Vichy-aligned during the War. This answer will focus primarily on Martinique, as it was the political center for the Vichy regime in the Caribbean.
At the conclusion of the Battle of France and the establishment of the Vichy collaborationist regime, Admiral Georges Robert, the colonial High Commissioner of the French West Indies and Guiana, declared allegiance to the Vichy government as the legitimate government of France. There was immediately great fear from the Western Allies that Martinique and Guadeloupe would be viable Nazi naval bases in the Caribbean as the Battle of the Atlantic began to hit a boiling point. For a time, New Orleans and Mobile were two of the most dangerous ports on the planet to ship goods out of due to the German U-boat presence in the Gulf of Mexico. Considering the volume of American ships sunk, sailors killed, and goods destroyed in the interim between the Fall of France and American entry into the war, Washington paid close attention to the new political developments playing out under the new pro-Vichy regime of Admiral Robert during the war.
While the US officially recognized Vichy as the legitimate French government at first, the military was acutely worried of the possibility of a full-on German takeover of the islands. Immediately, talk was done of a possible American invasion of the French West Indies island by island. It was however, unknown what the German response might be (if any). The War Department wondered if the invasions would precipitate a full-scale American entry into the war in Europe, or if occupying the islands and Guiana would be worth the economic or political resources that would be sunk into it. This debate was avoided in August 1940, when Vice Admiral John Greenslade and Admiral Robert signed the Robert-Greenslade Accords, a sort of gentlemen's agreement as author Kristen Childers put it. The deal entailed the US getting permission to set up a diplomatic consulate on Martinique (including naval personnel) and receive two days notice on the arrival and exit of ships to and from the French West Indies. The consulate would also be notified of political changes in the local Vichy regime, which they could then pass on to Washington.
In exchange, the US promised to allow the movement of food and fuel to the islands (rather than totally blockade them as the British government wanted) and not to directly interfere with the power of the local Vichy fascist government. The US would also sign the Destroyers for Bases agreement with Britain immediately after this deal with Robert, strengthening American presence in the Caribbean. The existence of this agreement to the American public was not revealed until almost a full year later in June 1941, when Secretary of State Cordell Hull discussed it in a widely publicized letter to Sen. James Mead of New York. There was some debate about this in Congress, with more pro-war voices detracting from the administration's secret agreement with a fascist state, and insisting on American occupation of the French colonies. Hull also revealed that the government had allowed previously frozen French funds in the United States into the Robert administration's coffers again as part of the agreement. This attracted political controversy, thought was soon overshadowed by the German invasion of the Soviet Union a few weeks later and the Japanese bombing of Pearl Harbor in December. German ships would soon begin docking at Martinique for fuel and resupply anyway, despite the presence of the American consulate. A share of American food aid and fuel supply to Martinique certainly ended up going to German naval personnel, who used those resources to go back out into the ocean to engage and destroy American ships and kill American merchant sailors. War is not without its ironies I suppose.
In Martinique, there was fierce resentment for the Vichy government. Vichy sailors and administrators were openly racist to the mainly black population, and the island took on a distinctly white supremacist character during the Vichy regime. American vice-consul on the island Harwood Blocker would write this:
"It appears that the general desire of the people in Martinique is to remain French. They do not approve of the Pétain Government, but feel that they have no other course open to them. Their friendliness to the British has cooled greatly, although there is no question about their desire to see the British win out in the present European struggle."
This white supremacist colonial pathology was not at all lost on a politically astute teenager by the name of Franz Fanon. Fanon's first ideas of the nature of European colonialism developed during his young adult life in Vichy Martinique, in which he said that the colonizers removed their masks and behaved like authentic racists. Additionally, the war totally disrupted the trade economy of Martinique, domestic crops like pineapple and sugar cane that would have normally been exported to mainland France rotted in the fields unharvested. Plantation owners and small farmers on island lost their incomes, and finished consumer goods that normally would have been imported to Martinique from France never came. This economic disruption also led the populous to detest Robert's government, and the relatively reliable stream of American foodstuffs fostered a continually pro-Allied consciousness among the populous.
This would eventually channel into civil protest and some forms of minor political resistance. Drivers would honk out the letter V in morse code to symbolize V for Victory in traffic. In at least one instance, football spectators left a match shouting "Bravo le Goal, Vive le Goal, Vive la France!", Vive le Goal being "Long live the Goal" and "Le Goal" sounding a hell of a lot like De Gaulle, the leader of the Free French Forces. There were political crackdowns by the government that involved some resistors getting sent to a prison camp in Guiana, but no too drastic actions could be taken out of the fear of American response. Thousands of people built makeshift rafts and left Martinique and Guadeloupe during this time, including Frantz Fanon. Many of these people, including Fanon, would fight in the Free French Forces for the remainder of the war. The United States took many of these anti-Vichy political dissidents. American officials were actually told to treat the mostly black dissidents with an uncommon sense of racial equality and to avoid segregating them, so as to not ruin the image of the United States as pro-freedom and pro-equality in the global fight against fascist tyranny. One man would later recollect:
"It seems that American authorities, conscious of such problems, tried to prevent Antilleans from experiencing racism. The Antilleans all recount that they were welcomed into white families and that from the moment they bore their epaulets marked ‘‘Free French,’’ they no longer had to worry .... On the whole, Antilleans still have very positive recollections of the United States."
In 1943, Free French Forces successfully liberated Guiana, and that summer, popular uprisings overthrew the Robert regime in Martinique and Guadeloupe. The 3 years of Vichy occupation were a very dynamic period for the French Caribbean colonies. It marked what can be considered the westernmost point of the Axis-aligned powers, an instance of Allied collaboration with fascist nation during the war, and as the historian Kristen Childers argues in her article on the subject, a watershed moment in which the black populations of Martinique and Guadeloupe genuinely developed a French nationalism as they overthrew their local Vichy state and declared the islands free French territory themselves.
Source:
Childers, Kristen. "The Second World War as a Watershed in the French Caribbean". Atlantic Studies (2012).