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):
While in the days of the Temple, the Avodah service was considered synonymous with the Yom Kippur experience, today our own cognitive association with Yom Kippur is that of a day devoted entirely to prayer. According to the Rav, prayer on Yom Kippur takes on a complexion fundamentally different from prayer during the rest of the year. The day of Yom Kippur must be transformed into a yom tefila, a day of prayer. To accomplish this transformation, Chazal instituted the Ne'ilah service. The purpose of Ne'ilah is to request that all the previous prayers of the day be accepted before God. (see Rambam, Hilchos Tefillah 1:7).
This conception of the role of the Ne'ilah service was so compelling to the Rav that he actually posited a halachah on this basis. If, during the year, one would forget to recite any of the three daily prayers (Shacharis, Minchah or Maariv) in its proper time, his halachic right to participate in subsequent prayers would be unaffected. If, however, for some reason one did not pray at all on Yom Kippur until the time for Ne'ilah had arrived, the Rav maintained that he could not participated in the Ne'ilah service. The function of Ne'ilah is to transform all previous prayers into one unified prayer activity. Without the earlier prayers there can be no Ne'ilah (Before Hashem, pp. 159-160). Elsewhere, the Rav suggested that one who missed even a single one of the previous Yom Kippur prayers may not recite Ne'ilah (Mesorah Journal, Volume 6, p. 23).
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:
כל תפלה שהיא נוספת מחמת מאורע היום אין ראוי להשלימה ביום אחר שכבר עבר המאורע.
Any prayer which is added because of a special day is not tashlumin-able because the special day has passed.
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.