What is the meaning of the phrase "recommended your firm without qualification?"
The meaning of this phrase is that whoever recommended your firm was 100% sure about their recommendation.
In this case, the word "qualification" does not refer to one's credentials that make one qualified for a job. Instead, the word is being used to mean something like "restrictions." If someone recommends your firm with qualifications, it would mean that they recommend you in some ways or in some circumstances but not in others. But since they have recommended you without qualifications, it means that they put no limits on their recommendation.
For example, a recommendation with qualifications might say that your firm would be good for small jobs but not for big ones. In contrast, a recommendation without qualifications would say that your firm was good for all sorts of jobs.