I didn't have any doubts about this word, because as I could see it is pronounced in both British and American variants as [ˈinfəntrē] - as it written - and I heard it in modern military usage sounding like that, until I heard an American WWII song The Ballad of Rodger Young in performance of West Point Cadet Glee Club - and it occurred to me, that they definitely pronounce it like 'enfantry' https://www.youtube.com/watch?v=1MEJM0cboDg#t=7 Then I searched some other variants of the song - and in the most of them the same pronunciation was used, although in some not. Then I have searched the Internet and have looked up in dictionaries, but nowhere this question was discussed. I think that maybe this is the variable pronunciation like insure/ensure, inquire/enquire, or dated, or dialectal, or specifically U.S. military, but still cannot find any arguments in favor of anything. Or is it just a personal accent of the singer and nothing interesting?
Choral performances (as in the case of a glee club) are not good sources for verifying issues of pronunciation, because it is commonplace in choral performances to alter pronunciation of words for musical reasons. One of the substitutions commonly made is to replace an initial "i" sound as you cite. The reason is that the "i" sound in "infantry" is pronounced with a small opening of the mouth, and as a result doesn't carry well in some rooms, while the "e" sound is pronounced with a much larger opening of the voice, and has more power and carries better. The short "i" sound is also much harder for an ensemble to tune as precisely than a short "e" sound.
In the case of the word "infantry", if one happened to be were listening very closely to a live performance, from the back of a large room (and perhaps one without very good acoustic characteristics to begin with), and heard a performance with "infantry" pronounced in with a short "i", the sound of the ensemble would seem to fade a bit on the "in" syllable while it would not if the vowel sound was modified as you noticed.
By the way, my credentials on this are that I sang in a professional quality ensemble for a number of years, and the substitution of short "e" vowel for a short "i" was such a regular occurrence that it was noted when we were not to do it. Not all choral conductors will make this modification, and some will make it only in some circumstances, for example, a particular ensemble, or a performance in a particular room. I would also note that this is a bit of choral technique, and may not (need to) be used by solo voices.