I don’t have a definitive answer, but I can hazard a guess. It seems to me like a case of false splitting. False splitting occurs in two ways, one is when a word originally had a vowel in the beginning, and the “an” that precedes it gets split so that the “n” attaches to the word. The other way is when the word originally began with an “n” and it eventually splits from the word and joins the “a” article that precedes it.

An example would be the word “orange,” as in the fruit. When Europeans first came across this fruit, is was called “norange” in its native language (different spelling, but same basic sound). So people would talk about “a norange.” Eventually, through usage, the phrase split into “an orange.”

My feeling is that this is what happened with “field trip.” To leave your usual place of residence or business is to go “afield” (often coupled with far, as in “we went far afield.”) If you were going to go on a trip afield, it is an easy transition to make to say you were going on an afield trip, or a field trip.

On the other hand, I could be all wrong. It’s a nice theory though!

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2 b : the sphere of practical operation outside a base (as a laboratory, office, or factory)

It’s not that the kids go out into an actual field. In this instance it’s used as an adjective to describe a trip. As a field of study – firehouse, pumpkin patch, bakery, etc.