LEOSA does not require you to qualify with a certain type of action. That is a limitation imposed by department.
Under the Federal Code that covers LEOSA, the definition for firearm “type” means either a handgun or long gun. This means, if you qualify with a handgun under LEOSA, you are covered by LEOSA for any handgun.
However, many states will break down “type” to mean either semi-auto or revolver. To break it down even more, some states will even go as far as different makes, such as my state of PA.
I’ve been doing LEOSA qualifications/certifications for quite a number of years. I asked for a clarification on “type” and the letter they sent me said, a person can qualify with a Glock, and be covered to carry any Glock. But, if they might also want to carry a Sig, because the “controls on a Sig are different from a Glock”, then that person would also have to qualify with the Sig.
And, if the might want to carry a revolver or snub, they need to qualify with that too.
I have to submit a form showing the make and model of the firearm used to qualify, but the card I issue them only has a check mark for semi-auto or revolver.
The NRA had a good FAQ on LEOSA and how states are making officers do things not required by LEOSA.