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)
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?