But the food delivered to the fortress was not for free ? Peasant don't deliver supply yearlong without cash.

In many cases they did, yes. For example, in 1813 many of the German allied states were sent demands for food and supply contributions for the French troops marching through en route to the new campaign in Saxony. These were never remunerated. In occupied Hannover the French "war contributions" were assigned a monetary value far in excess of what the locals could pay (the little village of Hameln, for example, was assigned 1.2 million francs), with the "remainder" being taken in food and supplies. (Usually listed as "Fournitures" on the French forms.)

The French even had the right to requisition people: if they needed a cobbler, carpenter, blacksmith, whatever, they could conscript him on the spot. I found records of this from Westphalia in the city archive of Braunschweig. Collection #: C.IV 9:16.

Either the french extracted money to buy food (and kept the change) or they received the food from local government,

I'm not aware of any examples of local state governments turning over food (it may have happened; I just don't know about it). Localities, however, Yes: the mayors, local religious leaders, etc, were often tasked with gathering the supplies for the French. You're familiar, I'm sure, with the great Stendhal (Henri Beyle), who was a green young lieutenant of dragoons in 1806, and who for two years had the job of cataloguing horses, food, vehicles, and other valuables in the region of Brunswick for confiscation by the French army. He interfaced with a local judge, who took him around and acted as his translator and made it "official."

The total French demands on Westphalia in 1808, for example, came to just under 50 million francs, or roughly 150% of that state's national annual budget. The records show that 24.8 million were collected in cash and a bit more than 25 million in "fournitures" (which usually actually meant: food, cloth, animals, leather, and other supplies.) This record is also in Berlin-Dahlem, Collection #: I. HA Rep., Nr. 157.

The cash contributions were collected by the local governments, yes. (Unless the regions were under French military occupation, such as Hannover, or Hessen-Kassel and Brunswick in 1806-7. In those cases the French administrators of the various "Pays Conquis" collected the cash and goods. Many of those forms have survived and are quite extensive. Sometimes an official Recette Générale des Contributions was written up, with amounts stated in francs. More often, however, a few sentences sufficed for a demand, usually in the local currency. I found one document, for example, in which two sentences sufficed for a requisition of 185,000 Thaler (about 721k francs), without even explaining what it was for, other than the French military. Sometimes, not always, an official would counter- sign the documents, taking responsibility for the contribution. It is striking, however, that so much wealth was extracted in this manner. It is also unclear, given the absence of official acknowledgment in so many cases, whether these sums were considered part of Napoleon’s official demands on these places (the "contributions"), or whether they were extracted in addition to those demands.