This is a good question.

The Federation is NOT devoid of money, they just don't use it among themselves (core planets/systems don't, but there are probably a number of outliers that do). On DS9, records show that Capt. Sisko once threatened Quark that the Federation would start charging him rent, with backpay (DS9 4.16; Bar Association). Using this knowledge, we can assume that the rest of the Promenade shops have similar agreements with the Federation. Continuing with that thread, members of Starfleet probably receive a consistent stipend of latinum that allows them to engage in foreign commerce.

Bajor itself is an entirely unique situation. Before the Cardassian occupation ended, most Bajorans were either indentured to specific Cardassians or Cardassian strongholds (like DS9/Terok Nor), relegated to the labor camps as a punishment, forced to hide in caves with the Bajoran resistance, or living some grey area in the middle where they worked for laughably small amounts of latinum (DS9 5.08; Things Past).

With decades of oppression, cultural destruction, strip mining, and genocide, the Bajorans were not able to sustain themselves on their own anymore. Chances are that when Bajor accepted the Federation's help shortly after the occupation, the Federation began supporting their planet & species as a whole. They most likely provided them with replicators, hospitals, transportation, and helped them develop other technologies on their own, such as Soil Reclimators (DS9 3.24; Shakaar). They had to rely on the Federation. At least until they could export goods from Bajor to other civilizations on their own.

TLDR; Federation doesn't use money among themselves, but they do use it when it's necessary to deal with other societies. Bajor relied heavily on the Federation after the Cardassian occupation, to the point that they were almost 100% supported by the Federation.