Could the infinitive phrase "to go" be a complement of the noun phrase "a coffee"?
Yes, I can't see a problem with that.
However, it is normally unnecessary - it makes no difference to the server if it is to go or you will drink it in the shop - it typically is "to go". If, however, it is a shop that offers ceramic cups for non-takeaway, you do need to specify.
Depends on your definition of complement. If you define a complement as anything that completes something then I guess you could call it a complement. I usually reserve complement for predicate nouns or adjectives, tho. I'd call this a post-modifier, ie, an adjectival phrase that occurs after the noun, similar to a prepositional phrase or a participial phrase.
a coffee to go
is a noun group with coffee as head element and "a" and "to go" as sub-elements (or modifiers or attributes).
To-infinitives can be used as modifiers: I'd like a book to read. One could derive such a structure by using three sentences: I'd like a book - what for? - to read it.
The derivation of the fixed expression of coffee shops "coffee to go" is a bit difficult: The sense is a cup of coffee that you can take with you (you don't drink it in the shop). That the expression uses "to go" is a bit funny, but it is short and understandable: coffee to go with you (out of the shop).
I wouldn't use the term complement, as it is ambiguous. Mostly complement is used in connection with linking verbs. In "He is a doctor" "a doctor" is a subject complement.
A coffee to go is like a table to eat. It's a necessary qualifier that indicates purpose, suitability or reason. It's necessary to avoid confusion or a mistake.
When I say: We would like that table to eat, it does not mean I want to eat the table. Neither do I wish for the table to start chewing on something. I'm asking for a menu and checking that the kitchen is open. If not, I will go somewhere else.
A "coffee to go" is a coffee that is for going (walking) and going (leaving) and going (not coming back). It is suitable to go, portable and disposable.
It doesn't always work: I bought this necklace to gift. We'd rather say that it is a present.
You go for a coffee or you go to drink coffee, but you do not say you go a coffee, so the phrase is a bit strange because of the collocations of the word coffee has no the verb go. A coffee to drink sounds better.