What you are describing is one of the functions of the helping verb (助動詞) てくる. Although the basic idea is the same: 来る/行く are a directional pair meaning "come" and "go", they have more than one figurative usages. The usage you are familiar with is expressions of temporal directions. In addition, てくる, ていく can also function to indicate spatial directions and conversational/interpersonal directions, of which your quoted line instantiates the latter.
Here in「言ってくる」「てくる」 tells you the direction of the action. It is a conversational/interpersonal direction. The speaker uses 言ってくる to indicate that they are at the figurative center and 「ワクチンを受けたい」may be said to them in an incoming and approaching direction. As an example:
A similar structure exists in English (and some other languages) as well.
If you have any questions come talk to me after class.
Here the "come" is both spatially and conversationally directional at the same time. The listeners are encouraged to talk to the speaker after class, but in order to do that they might need to physically approach the speaker first so that neither party needs to shout.
I am gonna go see him tomorrow.
"Go" here demonstrates a spatial movement away from where the speaker's perspective is anchored.
The roof came crashing down.
Both physical and personal directions.
My life came crashing down.
Personal and emotional directions.
So your sentence means:
If someone who does not have family registry documentation comes to the public office and says they want to get vaccinated, the local city or town office will check if they really live there and send that public office a vaccination coupon.