How are things purchased in AoT?
There's definitely a monetary system in place (note I'm on mobile so I can't link any sources yet).

In episode 1, after the incident where Moses's arm is handed to his mother, there is a person saying "its like our taxes go to fatten them up". This clearly indicates some form of tax system.

In episode 6, the titans yet again breach another wall. There's a scene where merchants are blocking the passageway trying to get their goods through. This shows an obvious ownership of goods.

In the same episode, we see Mikasa and her mom targeted by slave traders. The slave traders planned to sell them to some rich folk, demonstrating value and a need for some form of money.

In episode 23, we can clearly see corrupt military police selling goods for what is obviously currency.

Over all we see several trends that point to the use of currency within Attack on Titan.