Three reasons, all very straight forward. The first is that the Vichy state was certainly subservient to Nazi Germany, but Vichy France nevertheless retained basic autonomy, and its overseas territories were officially neutral like Vichy France was. They were obligated to keep the Allies out, which resulted for instance in brief combat between Vichy and Allies forces in North Africa and Syria, but not expected to participate as a belligerent, and the flip-side was that the Germans couldn't make free use of French territory either. They did eventually move to occupy all of France, but that is another matter.

The second is that the Nazis didn't possess teleportation technology. Even if they could surmount the first issue, and get the agreement of the French government to allow offensive operations to be conducted from there, they would still need to get there. Aside from their U-Boat fleet, which certainly would not provide enough of a logistical backbone to provide meaningful support to forces based from there, the Kriegsmarine absolutely did not have the ability to reach French Guiana in any meaningful way, as demonstrated by the fate of the Bismarck.

Finally it is also worth noting that even if the Germans could get there... it would have been a pretty tough place to operate from. It had a tiny population of 30,000, 6,000 of whom were convicts, and lacked much in the way of infrastructure. Even building into a place capable of supporting meaningful operations would have been a major enterprise. It had no meaningful defensive capabilities of its own, and the US was keeping an eye on things even before their entry into the war. If there was a hint of utilization by Germany, the Allies would have moved into occupy it quite immediately.

Baptiste, Fitzroy A. "THE ANTI-VICHYITE MOVEMENT IN FRENCH GUIANA, JUNE TO DECEMBER 1940." Social and Economic Studies 26, no. 3 (1977): 294-307.

Before Germany can even attempt to get to the French colonies it needs to ferry its troops, supplies, and other material across the ocean. And it needs a large fleet and a large amount of transport ships as well. However Germany’s fleet was nonexistent and wouldn’t be able to guard a transport fleet.

By the time of the Fall of France, Germany’s fleet was devastated after the invasion of Norway. It lost a large amount of cruisers, destroyers, subs, and transport ships. But to rebuild these ships Germany needs time and resources something Germany doesn’t have. It’s takes months or years to build ships and maintain a fleet and Germany had a handicap due to the Versailles treaties and its own limitations. And remember every ton of steel going to ships is not steel going towards tanks or trucks. Fleets also require a lot of oil to move the ships, Germany has an oil shortage already and this would exasperate the problem. It was barely getting enough as is, a large fleet would not be possible. And there was no way the British were going to just thumb their noses at this as well. They are the dominant naval power in the area and no one else could match it.

And this bring me to my next point of logistics and planning. The British aren’t going to allow a large German fleet to leave the North Atlantic or the coast of France. It would notice any significant build up as a threat of invading Britain. Germany had troubles planing a landing of a invasion force of 50,000 on British soil across the English Channel, a 3,000 mile trek across hostile waters to South America would be suicide. German planners had troubles planning a supply effort to the Germans even if they somehow successfully landed on British soil. Those troops even if they got there would be stranded and cut off from all supplies. To naval invade anywhere, you need to keep the troops well supplied, well supported, and with new troops landing everyday to reinforce the beachhead.

But the question you need to ask is if Germany even needed to move troops to Guyana? The answer is no they didn’t. They had no reason to, they already had enough problems with the UK and the upcoming invasion of Russia. Why would it send any troops to French Guyana when it can’t support any troops sent there, even if they did they would be intercepted by the British fleet and destroyed, and they’d waste men and supplies for no gains. Germany can’t waste time and resources like that it needs to fix its oil shortage soon and Russia who looked weak after its fiasco in Finland looks like easy pickings. And the Russians had oil, large amounts of fertile land, other minerals and natural resources Germany needed for its war machine.

In the end Germany had no possible way of reaching the far flung colonies nor did it have really any desire. And there was no real benefit of them landing troops there or establishing a large presence as it the cost outweighs any benefit.