Why is there no 'e' in this sentence to specify what he is training for?
-'e' is a noun suffix and is only required when you are using a pronoun-as-verb "to be" construction. If he was training for a thing (a battle, a test, an emergency, etc.) then there are ways that -'e' could be used on the noun (though it still wouldn't be required). But in this case he's training to do an action. The verb suffix -meH is being used here to indicate the action that he is training for.
It has just occurred to me that you might mean the pronoun 'e', rather than the suffix -'e'. That would be an acceptable way to say it as well. mang mojmeH qeqtaH mangHom has a feeling like, "The cadet is training (in appropriate things) so that he can become a soldier." mang moj 'e' qeqtaH mangHom has a feeling more like, "The cadet is practicing becoming a soldier." If I think philosophically about it, the sentence with 'e' sounds more like he is practicing the graduation ceremony rather than the skills needed to qualify. But perhaps that's thinking too hard about it and I could defintiely see that sentence being reasonably used to mean the same as the sentence given. I still think the -meH version is better, but I'll add the 'e' version as a possible translation of the English sentence.