Where does Natsu and his party get money from, to pay their living cost? I started watching Fairy Tail and have just finished the Galuna Island arc where, once again, the reward is turned down for pointless "moral" reasons.

Unlike One Piece (where they can just steal/procure food and such), they do actually require money for paying rent and such.

Do we ever actually see Natsu's party claim a reward? Otherwise, where do they get their money from to pay their living cost?
As Aki Tanaka has recommended. I have expanded my previous comment as an official answer. Fairy Tail is guild noted pre-time skip and post-time skip for two things:


Blowing s*** up
Getting the job done


The guild hall has been destroyed completely on more than one occasion along with towns and villages being blown to the ground due to battles, scuffles and even arguements amongst members. This is shown early on from Natsu v Gray, Natsu v Erza, and pretty much Natsu v anything. 

It's shown in a few episodes that Natsu and the team take a job and either don't feel comfortable taking the reward because it got done in a round about way (like in Lucy's first job) or the money that they were supposed to get winds up getting donated by the guild to rebuild what ends up getting destroyed.

Now to answer the question as to how these guys even survive, this comes from two things.

The first is pretty obvious which is that not all of their members outright end up destroying stuff, some of their top-tier wizards (Gildarts,Erza, Mystogen, Lauxus, etc) do take jobs that pay lucratively high and bring home the bacon which happens to cover it.

The second is up to speculation and there are some theories out there. But the cannonical answer is that the plot demands it. The heros are supremely powerful but kept in check constantly by the laws of the society in which they reside. Kind of like how Luffy and the strawhats are rarely ever shown being stocked up on ship supplies.