"strengthen" doesn't quite work here, in my opinion as a native English speaker (from USA). "Strengthen" takes a direct object, so you could say "the soldiers strengthen the base's defenses" but it doesn't work to say that"they strengthen themselves" or that they strengthen".
I have translated "מתאמנים" as "work out". Is it not an accepted translation in this context?
Very nearly. If there's a difference I'd say that becoming stronger feels a bit more immediate, whereas getting strong feels like it might be a bit more in the future, but that could be just me.
Is הוא מתאמן להתחזק a valid sentence? I usually check things like that in google to see if some phrase appears on the Internet in a natural context and there are no results for this one - so is this unnatural or something?
As in "He trains to get stronger"? Not really because starting the second verb with a lamed makes it an infinitive, not the lamed that translates into "to".
I would say "הוא מתאמן כדי להתחזק" which makes it clear that we are talking about purpose here.
Can you translate מתחזק as "become stronger"?
To me, it has the same meaning as "get stronger" and is less colloquial.