And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth. - John 1:14 KJV
The word in question is γίνομαι (ginomai) and Strong's Concordance defines it's usage as:
A prolonged and middle form of a primary verb; to cause to be (“gen” -erate), that is, (reflexively) to become (come into being), used with great latitude (literally, figuratively, intensively, etc.): - arise be assembled, be (come, -fall, -have self), be brought (to pass), (be) come (to pass), continue, be divided, be done, draw, be ended, fall, be finished, follow, be found, be fulfilled, + God forbid, grow, happen, have, be kept, be made, be married, be ordained to be, partake, pass, be performed, be published, require, seem, be showed, X soon as it was, sound, be taken, be turned, use, wax, will, would, be wrought.
The verb is most often translated as "became" or "was made" with the occasional "came in" or "took on". In John 1:14 the voice of this verb is middle deponent:
The word εγενετο is the 3rd person single form of the verb marked similar below. Its tense is 2nd aorist (which indicates the mere fact of the action, with deliberate silence about when the action takes place or how long it would last), its voice is middle deponent (which indicates that the subject performs the action, instead of receives it), and its mood is indicative (which describes a situation that actually is — as opposed to a situation that might be, is wished for, or is commanded to be). - https://www.abarim-publications.com/DictionaryG/g/g-i-n-o-m-a-i.html
An online greek course has this to say about middle voice and deponent verbs:
When the middle voice is used the subject is performing the action, but is also involved in some further way in that action. The subject may both perform and receive the action, or may perform the action on his own behalf, or in some other way be more intimately involved in the action.
Many verbs have middle or passive forms that will often be translated into English using active voice verbs. An example is ἔρχομαι, I come, I go. Traditionally, these have been called deponent verbs. The word deponent is from the Latin deponere = to lay aside. This term suggests that the middle or passive meaning was laid aside for these particular verbs even though the middle or passive form was used. However, for many of these so-called deponent verbs, it may well be that the Greek speaker really had a perspective on the action that made a middle voice appropriate, even though in modern English we would tend to describe the action using an active voice.
Since the Word is the subject of the middle deponent verb (which indicates that the subject performs the action), is it fair to say that the Word performs the enfleshment rather than being enfleshed by another?
Note: Apologies for the creation of the word "enfleshment". Oddly enough, spell-check recognized "enfleshment" but not "enfleshed" :-)