Reinforcement Learning with static state Can Q Learning work with a static state for each step? 

What I mean by that is that the actions do not influence the following state at all. The episodes just iterate over the same data over and over again. Of course the different actions lead to different rewards, but is Q Learning the right concept to use in this case? What other types could you suggest?
The framework you describe is the Bandits framework. Bandits are algorithms that solve stateless MDPs. So the problem is not sequential in the sense that actions affect the transition to the next state. Take a look at a nice Contextual Bandits tutorial which is accompanied with Python Jupyter notebook. Please note that Contextual Bandits are a special category of Bandits but at least you will be exposed to the general Bandit idea and formalization.

You can try Q learning but with the Rescorla-Wagner rule $r(a) - Q(a)$ as the reward correction $\gamma\max_a' Q(s',a')$ does not make any sense as there is no transition to a subsequent state $s'$.