What happens to the people who worked for the Inquisition if it is disbanded?

I suppose it depends on what the understanding was when they were originally recruited. Did Cassandra/Leliana or their subordinates tell them it was permanent or just temporary during the current crisis? I'd assume the latter because everyone would be hoping that things would settle down, a new Divine would be elected and she would then decide things going forward. As the new Divine was elected from the ranks of the Inquisition it would seem that she decided to leave it running even after Corypheus had been dealt with and that probably gave it a more permanent feel since people who had been there from the outset had been in post for the best part of nearly 4 years by the time of the Exalted Council. Even if it continues as the Divine's private army, it is still considerably cut down in size, so most of the auxiliary positions would go, plus a fair proportion of the armed forces. However, being able to give your last job as a member of the Inquisition probably amounted to a positive recommendation to any future employers and they had had the benefit of steady employment at a time when many people were displaced refugees from various wars, so that should have given them sufficient funds to tide them over during the time it took them to find jobs elsewhere.

Given the number of civilians that had died on the Exalted Plains and various locations in Ferelden, I should imagine that helping to restore the economy in those regions would still be a work in progress and initially work probably wouldn't be that hard to find.