What would be the gender-neutral version of the term "dairyman"?
I would suggest "dairy farmer" if you mean the owner of the dairy farm. And "dairy farm worker" if you mean the general worker. These are the official occupations (at least in most of the English speaking countries)
Also, the business is called dairy farming.
There are even dairy farmers associations in different countries:
Here is a list of dairy occupations (from the book "Modern Livestock & Poultry Production" By James Gillespie, Frank Flanders):
How about dairy worker or dairy keeper, depending upon the employment status of the individual concerned?
Depending on context, consider:
and dairy operator