Filth flies, carcass inhabiting flies and even some associated with pollination biology, have been related to disease, allergy and myiasis...... but how is it they are pinned to their respective diseases and allegries.
There are many methods, including PCR, direct washing and elution followed by centrifugation and acid-fast staining, and others. In our lab, we use several methods on muscid, sarcophagid, and calliphorid flies. Our primary method has been multiplexed fluorescent in-situ hybridization (FISH) combined with immunofluorescent antibody (IFA). I have listed below 2 links to files related to this work. Let me know if you want additional references from our work. We have worked more with molluscs, but the items below reference flies in terrestrial ecosystems.
Best wishes,
Bruce
Article Synanthropic Flies as Vectors of Cryptosporidium and Giardia...
Article Biomonitoring for zoonotic protists in terrestrial and aquat...
For intestinal coccidia, I did it microscopically; but it was time-consuming work. REFERENCE: Markus MB. 1980. Flies as natural transport hosts of Sarcocystis and other coccidia. Journal of Parasitology 66 (2): 361-362.
I would say, the first step is to detect your pathogen in the fly (or on the mouthparts for instance if you are just looking for mechanical vectors by biting). You can use for that some molecular methods like PCR or other ones (FISH,...), or observations of the pathogen by another way like suggested. But it stay just a possibility that this fly is a mechanical vector of the pathogen observed, and don't demonstrate it is really one with an epidemiological value. It stay hypothetical.
Then it become more complex and it depends of the model of transmission and the pathogen studied. As you already show that the fly can harbor the pathogen in natural condition, you now have to show it could lead to mechanical transmission. For that you could experimentally demonstrate the transmission in an experimental model or try to demonstrate it really happens sometimes in natural conditions (by different direct or indirect ways). It is really depending of many things at this point.
For the allergies, it is totally different, you have to show that at least one fly protein or some mix of proteins are allergenic to the mammals studied. It could be done by fly protein purification and some in vitro immunological technics with blood or purified blood cells from allergic mammals... and it is more complex than just a PCR...
Your question is more complicated than it seems. To really prove a vector you would need to show that the pathogens are present in them, actually spread or transmitted to a host, disease occurs, and it is the same organisms that were in the fly.
The association between flies and disease is common and based on antecedent evidence but it is a really hard thing to prove. It is an ongoing issue for vector biologists and just because you find a pathogen in a fly it does not make it a vector.