In Middle Earth, dwarves and men had a long history of forming mutually beneficial economic relationships (which you can read all about in this post.) However, this was when both parties had stable and peaceful settlements. The situation facing the dwarves of Erebor after Smaug’s attack would have been very different.
The years between the destruction of Erebor and the War of the Dwarves and Orcs are described very vaguely by Tolkien as a time of “long and homeless wandering”, generally in the region of Dunland. Dunland is, primarily, inhabited by a culture of men called the Dunlendings. It’s not said what sort of relationship the dwarves and Dunlendings had, but we do know, at least, that Thror wasn’t happy with it. When he’s preparing to leave (unknowingly traveling to his death in Moria), he tells his son Thrain that he was “tired of poverty and the scorn of Men.” It could very well be that this was because the dwarves were being paid little to do what they considered to be menial and humbling work (especially for the more skilled craftsmen among them.)
After the Battle of Azanulbizar and the end of the War of the Dwarves and Orcs, Thrain and Thorin have a conversation about their next plan for their people:
Then standing by the great stake, Thráin said to Thorin Oakenshield: ‘Some would think this head dearly bought! At least we have given our kingdom for it. Will you come with me back to the anvil? Or will you beg your bread at proud doors?’
'To the anvil,' answered Thorin. 'The hammer will at least keep the arms strong, until they can wield sharper tools again.'
So Thráin and Thorin with what remained of their following (among whom were Balin and Glóin) returned to Dunland, and soon afterwards they removed and wandered in Eriador, until at last they made a home in exile in the east of the Ered Luin beyond the Lune. Of iron were most of the things that they forged in those days, but they prospered after a fashion, and their numbers slowly increased.
This conversation is interesting because it shows a completely different attitude towards their situation than Thror had. While he was concerned with poverty and “the scorn of Men”, Thrain and Thorin seem to think that their humble independence was better than living in the debt of their kin.
So, while Tolkien never says directly that dwarves were working for men during this period, we have to assume that that’s how they survived. There’s such a long tradition of dwarves and men relying on each other economically, that I can’t imagine this would change much after Smaug’s attack. The main difference would just be that their humble situation would be a blow to the dwarves’ pride (or, at least, it was to Thror’s.)
SOURCES: LOTR Appendices