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R' Joseph B. Soloveitchik's understanding of Ne'ila, which I saw in the Machzor Mesoras Harav, is that it's a uniquely dependent prayer whose purpose is to ask God to accept all the other prayers we've engaged in over Yom Kippur. He was confident enough in this understanding that he proposed a practical Halachic outcome: If someone happened to miss all four of the preceding Yom Kippur prayers, that person would not be allowed to daven Ne'ila.
Based on that, my guess would be that Ne'ila is different enough from other prayers that indeed, it can't make up for them or be made up for.
If I'm right, the additional question would be whether the preceding Yom Kippur prayer, Mincha, could be made up for during the first Ma'ariv after Yom Kippur.
UPDATE: Here's the language from the Machzor (p. 768 in the first Ashkenaz edition; some Hebrew transliterated by me):
The first citation is simply to a previous work by the Machzor commentary's author in which this piece of commentary previously appeared. The second is quoted in a footnote in the former, reproduced here with thanks to Amazon:
The last position would certainly be incompatible with making up for Mincha at Ne'ila, and I maintain that the overall message is incompatible with Ne'ila being made up for by any other prayer.
Rav Ovadia Yosef has a teshuva (Yabia Omer OC 7:54) on the former question (making up for a missed N'ila) dated 11 Tishrei 5748.
He quotes Tosfot (Brachot 26a sv Iba'y) who gives two reasons that there is no tashlumin for a missed Musaf: because you can't say the verses related to the korbanot on the wrong day, and because Musaf was only established to take the place of the Korban Musaf whose time has already passed. Since neither of these reasons applies to N'ila, it would seem there would be tashlumin for a missed N'ila. The Peri Megadim in fact rules this way (OC 108 MZ 5) although he doesn't quote earlier sources for his ruling. (This leads to the strange case of praying thrice: Maariv, Tashlumin for N'ila and Tashlumin for Mincha.)
However, the Rashba (Shu"t 1:447) explains that there isn't tashlumin for Mussaf because:
According to this reasoning, N'ila would not have Tashlumin. Rav Ovadia quotes a slew of Rishonim who also give this reason, among them Ritva, Meiri, Tashbetz, Ra'ah and Rif and he seems to conclude in this direction as well.
Incorporating some ideas that were mentioned in @IsaacMoses's and @DoubleAA's good answers, this question was asked of R' Ephraim Greenblatt and he printed the answer, YES, in a collection of short piskei halacha (legal decisions) called Riv'vos V'yovalos. Paragraph 73 there says that although the P'ri M'gadim in Mishb'tzos Zahav (108:5) doesn't give a reason for allowing compensatory ma'ariv after missing n'ila, his opinion is essentially sufficient to base practical advice.
There he also makes reference to his response to the question about missing mincha on Yom Kipur and making up for it upon remembering after nightfall by repeating ma'ariv (Riv'vos Efrayim I:417). He concludes that despite the rationale comparing n'ila to musaf in its critical placement in the schedule, they are not actually identical so the rule applying to musaf need not necessarily apply to n'ila as well, and the opinion of P'ri M'gadim can be taken at its word.