I assume ご連絡申しあげます and ご連絡を申しあげます have the same meaning, but differ in style. I use them interchangeably below. This question is not about existence of を unless it was relevant to understanding the phrase.
Having received an email ending with announcement of future contact in the form of 「ご連絡を申し上げます」 I am wondering how this phrase is built up, so that it is accepted.
My main point of confusion is using the verb 申し上げる a humble form of "to talk", "to say", "to express" with a noun 連絡 meaning "connection", "contact", "communication". Using both words as predicate and subject makes no sense in English and probably most Indo-European languages (while the same words have equivalent usage patterns in many other combinations).
To help understand readers, what is my concern, I have following hypotheses:
1) Is 連絡 a thing that you can say or express like お礼を申し上げます/お礼を言います? Can I simplify (keigo aside) it to 連絡を言います? (I can't find any proof in dictionaries it works this way)
2) Maybe as suggested here or here 申し上げます preceded by お〜/ご〜 works like 致します, but how universal would it be? Can I say お待たせを申し上げます?
3) If お待たせ example does not sound strange, maybe the construction is some kind of a contraction of (お待たせいたしました)を申し上げます being "I would like to express - sorry to keep you waiting".
4) Or maybe it is just a set phrase ご連絡申しあげます coming from custom or tradition?