That information is not necessarily public information I suspect, and may not be the same in all vaccines.
There has been some discussion of varying doses of antigen in the AstraZeneca Oxford vacccine - because some of the first injections were a "full dose" ( without specifying what a full dose actually is ), and later they admitted that SOME, but not all, of the patients first dose was only a 1/2 dose. All patients then got a "full dose" for the second shot, - but apparently the folks who got a 1/2 dose and then a full dose, had higher levels of antibodies - a result which seems a bit unexpected, confounding even. And may need further study, since they subsequently revealed that the 1/2 dose first shots were done in error, were not a planned protocol.
Not terribly reassuring news from a vaccine manufacturer, that their study contains known administration errors.
How the amount of antigen is chosen to inject in subjects is not made public - but I suspect is either derived indirectly from animal studies, or from previous vaccination research efforts.
One doesn't want so much antigen it makes subjects ill - fever, malaise, myalgias, serum sickness - but does want enough to stimulate the immune system. I suspect there are known standards from previous vaccination efforts ( with different, but similar, bacterial or viral antigens ) from which to begin.
You gotta start somewhere, so start about where things worked before - so to speak. If necessary, then makes gradual changes. Or, like a cook would say, salt to taste.