The presence of a moderator would mean that some process proceeds differently in two separate groups. This is basically a hypothesis that could be tracked by coding the qualitative data to match the variables you have in your hypothesis.
But note that my use of the words "variables" and "hypothesis" indicates that this is a theory-driven form of analysis, which would not be compatible with some approaches to qualitative research, such as Grounded Theory. In essence, it would involve what is known as deductive rather than inductive form of content analysis.
My view is that this type of study would be ideal for a qualitative investigation. Essentially, if you want to know about the effects of a moderator on existing relationships, semi-structured interviews wold be a good way of going about this. I have used grounded theory to explore patients perceptions of GPs, chronic disease patients perceptions of various physical activity interventions etc. What I have been doing is interviewing multiple patients to gain the perceptions of the same thing i.e. physical activity interventions. While my studies are more in the medical domain, you could use the same techniques to gain insights into the impact of the moderator on the 'existing relationships'.
I have attached one of my papers that demonstrates the process that I used.
In an explanatory sequential design, you would begin with a quantitative test of the moderating hypothesis, and then follow-up with a qualitative study to improve your understanding of those results. Is that what you have in mind?
Since you are working with organizations, then you could use to quantitative results as a basis for purposively selecting a set of cases to examine in your follow-up qualitative study. Which cases you would want to study would depend on the results of quantitative analyses.