Is there any shorter way to say "military medical personnel"?
I mean by that: all the people in the armed forces that are allowed to use medical equipment on a daily basis.
In the military, personnel assigned to the medical field are often called medics. So, if enough context was provided – that is, if it was obvious we were talking about military personnel – you could use that word.
Alternatively, you could use the word corpsman, but that word is reserved for enlisted personnel, so that word would exclude military doctors.
These words aren't perfect, though. As you can see, medic is sometimes restrained to combat corpsman. From an online dictionary:
With that in mind, it might be better to use the longer form you provide:
This would include, presumably, all military doctors, pharmacists, etc. – people who might be assumed excluded if the term medic or corpsman were used.
You should clarify your question. You mentioned "military medical personnel" which would mean medical personnel of armed forces in general. But, you also used the word army, which would mean just medical personnel in an army and not, medical personnel of armed forces in general.
In the general sense, the words "military medical personnel" would be fine. In the particular example of medical personnel in an army, the words "army medical personnel" would be fine.
Sometimes, people say "medics" as an informal way to refer to medical personnel. In a military context, "medics" means enlisted medical personnel in particular.