Why take "will" in "Many people will go to the station every morning" as complement to the verb phrase "go to the station every morning"?
For some reason I read this question differently. I thought that you originally wanted to ask: Why take OUT "will" in the sentence "Many people will go to the station every morning" as complement to the verb phrase "go to the station every morning".
The answer would be that you do not need to take it out because, as the previous answers accurately stated, "will" is not a complement, but it is actually a verbal auxiliary and a transitive verb.
If you are asking what would happen if you take "will" out of the sentence, within an ESL context, the answer would be that the sentence will change meaning completely, since "will" entails (like the previous answer stated) that there is a predictability as to what will happen every morning at the station.
Another thing that could happen if you take out "will" from the sentence, is that it will change tenses. It will go from future to present, and so the entire meaning will change, as well.
The helping verb "will" is not a complement "to go to the station everyday," or vice versa.
"Will go" is the verb phrase. In English the tense is often signified by a helping verb, such as in this case. Another example is the pluperfect tense, when we use the helping verb "had." "She had gone to the station everyday."
"To go to the station everyday" is a prepositional phrase. It is used adverbially to show where the person is going. It is akin to saying that she is going "there." "There" in the sentence would be an adverb.
So in your sentence, there is actually no verb complement. However, if you have to have one, then the only phrase that can be a complement is the prepositional phrase "to the station everyday."
"Will" makes the future tense out of the verb "go." The speaker is predicting that sometime in the future many people will travel to the station, based on previous experience implied by the adverbial phrase "every morning." The construction is an example of how the English language has a propensity to denote tenses with auxiliary verbs rather than inflections or changes in spelling. The term "complement" is not perfectly accurate here, but without a thorough explanation of the grammatical variations available to the English speaker, it will have to do. Any good grammar book will have a chapter on verbs, to which I recommend you.
"Will" is not a complement; it's an auxiliary verb, also called a helping verb, with "go" as the main verb. We don't call verbs complements; generally, a complement is a noun that follows a verb & renames the subject, as in "John is a student," where "student" follows the verb & renames the subject, or an adjective that follows the verb & describes the subject, like "John is smart." We call these adjectives & nouns linking verb complements, because the verb links the subject with the noun or adjective that follows it.