The chief reason to consider it as a unit would be to add it to a list of prepositional verbs to be memorized by learners. In that case rely on would make a tidy and useful addition. Its status as a syntactic constituent, however, does not stand up to scrutiny.
According to the Oxford Dictionary of English Grammar, a constituent is:
A linguistic unit that forms part of a larger structure.
What is the larger structure? A verb phrase.
What is at the head of a verb phrase? A verb.
What is the most readily identifiable property of a verb? Its inflections!
Which inflects: rely or rely on? Only rely.
Contrast this with the verb one-up which does inflect as a unit one-ups, one-upping, one-upped.
Further, note that on may always be separated from rely by a modifier.
Mary relies solely on John.
It's John that Mary relies solely on.
John, Mary relies exclusively on to do her shopping.
Thus, the other constituents (the required complement headed by on and any possible modifiers) of the verb phrase headed by rely are clearly separate from it.
Supposing one seeks a further reason not to consider rely on as a constituent, if it were considered a constituent there'd also be the need for a unit intermediate between verb and verb phrase that would have to be created in order to account for 'the verb and the word heading its required complement, but excluding any dependents in that complement' - an awfully specific and entirely unnecessary unit.
The analysis that works in all situations quite cleanly is clearly the one to be preferred: Rely is the head (constituent A) of the verb phrase. On and any dependents thereof form a complement (constituent B) in the verb phrase headed by rely. The complement headed by on is a required one (it may also be headed by upon, whereon, thereon etc.).