The gender of the part would not take precedence over the gender of the whole when using using “Vatican” and “Church” as components of a synecdoche.
The church as a whole is female; it’s the Bride of Christ, as noted above.
Example from an old hymn:
“The Church’s one foundation is Jesus Christ her Lord;
She is the new Creation by water and the Word.
From heaven He came and sought her
To be His holy bride
With His own blood he bought her
And for her life He died.” Samuel Stone, 1866
That’s not quite correct. German is an Indo-European language with three genders, so it doesn’t lack an “it”. Things can be in any of these three genders, for example:
der Löffel, m, the spoon
das Messer, n, the knife
die Gabel, f, the fork
Perhaps they only emphasize it when they’re being blamed for something.
But is neuter really considered a gender? Or lack of a gender?
Yes, it’s a grammatical gender, like masculine and feminine. In Dutch and Swedish, if I have this right, they have two genders: common, which corresponds to masculine and feminine in other Germanic languages, and neuter.
In non-Indo-European languages with lots of different genders often without an equivalent to masculine and feminine, the more general term “noun class” is sometimes used.
Like Indo-European languages, classification is often arbitrary; for example, in Ojibwe, nouns are divided between animate (living and sacred things) and inanimate (other things), much as most things that are actually male and female are masculine and feminine respectively in French, but there is some arbitrariness; for example, “raspberry” is animate but “strawberry” is inanimate.
For another example, Luganda has ten noun classes, which can be summarized as people, long objects, animals, inanimate objects, large things and liquids, small things and abstracts and negatives, languages, pejoratives, infinitives, and mass nouns; but most of these classes have all kinds of random nouns in them that have nothing to do with the preponderant members of the class.
An encyclical of Pope Pius IX, “Cum Sancta Mater Ecclesia” (translated “Now when Holy Mother Church”) Cum Sancta Mater Ecclesia - Papal Encyclicals
“INSTRUCTION CONCERNING THE HISTORICAL TRUTH OF THE GOSPELS” (1964), translated begins “Holy Mother the Church” http://catholic-resources.org/ChurchDocs/PBC_HistTruthFitzmyer.htm#PBCText
And John XXIII’s social justice encyclical was called “Mater et Magistra” and started “Mother and Teacher of all nations—such is the Catholic Church in the mind of her Founder, Jesus Christ. . .”
Yes, and she’s hitting on menopause right about now, so forgive her for the current bout of crankiness.
I think the point raised was not that all Indo-European languages lack a class for a neuter noun, but that English has no classes of the “gender” type – prevalent in Indo-European – at all. Since most Indo-European languages use two classes, male and female, all their nouns, and more importantly their pronouns, will in translation acquire a gender that they would not have in the original English.
The other point raised by the original post was that not all language families classify by gender, but some use different methods to classify their words.
In other words, Paul wasn’t saying German wasn’t Indo-European, or that all Indo-European languages lack a neuter; just that most languages of the family make do with two classes.
In any case, yes: “The Church” (the body of believers and organization of clergy) and “The Holy See” (the legally-sovereign entity at its head), as nominal phrases take the feminine gender by the rules of Latin and its daughter languages, since the root words are feminine-gendered. When translating into English, the translator will often retain that affectation.
The grammatical gender attached to the “State of Vatican City” OTOH can vary (Spanish, for instance, would use masculine gender for “el Estado de la Ciudad del Vaticano” or “El Vaticano”, yet use feminine if saying “la Ciudad del Vaticano”. BUT, we normally use the latter by itself only to refer to the geographic location, not the theocratic body politic, for which we use the former.)
BTW if it’s not yet clear from prior discussions, assigning grammatical gender to inanimate or abstract nouns in certain languages does NOT mean that it is claimed the thing being named necessarily is somehow existentially “male” or “female”.
This point needs to be emphasized; grammatical gender is only loosely related to personal gender once we move beyond words that describe males and females.
Hell, mentula–a Latin slang word for the penis–is grammatically feminine, while venter - “womb” is masculine.
Not always. More often, it seems, vagina ensues.
Never argue with Chronos. My new motto 
Thank you all for a completely enlightening answer to my query. I’d no idea that it was such a well-defined identifier in the Church.