This answer address the question in a general way, regarding a man who has no legs. it treats the question if someone cannot make Chalitsa is there an impossibility to make Ybum as someone who cannot make Ybum cannot make a Chalitsa.
The known rule in Gemara Yevamot 3a is:
NOT Ybum => NOT Chalitsa.
Yevamot 3a:
ליתני מן הייבום לחודיה אי תנא מן הייבום, הוה אמינא מיחלץ חלצה, יבומי לא מייבמה, קמ"ל: כל העולה לייבום - עולה לחליצה, וכל שאינו עולה לייבום - אינו עולה לחליצה
What justification is there for stating, FROM THE HALIZAH AND FROM THE LEVIRATE MARRIAGE when it would have been sufficient to state FROM THE LEVIRATE MARRIAGE only? - If FROM THE LEVIRATE MARRIAGE only had been stated it might have been assumed that she must perform halizah though she is exempt from the levirate marriage, hence it was taught that whoever is subject to the obligation of levirate marriage is also subject to halizah and whosoever is not subject to the obligation of the levirate marriage is not subject to halizah.
Case of a Yabam without legs and possibilities of Chalitsa
Mishna Yebamot 12, 1:
מִן הָאַרְכֻּבָּה וּלְמַטָּה, חֲלִיצָתָהּ כְּשֵׁרָה. מִן הָאַרְכֻּבָּה וּלְמַעְלָה, חֲלִיצָתָהּ פְּסוּלָה:
Below the knee [the halizah is] valid, but if above the knee it is invalid.
The Gemara learned this from verses (103a):
below the knee etc. a contradiction was pointed out: regalim, excludes stump-legged cripples! - Here it is different since it was written in scripture, from off hi foot. if so, [halizah should be permissible] above the knee also! - from off but not 'from off the from off'.
Thus, some Yebamim are unable to fulfill their role in Chalitsa ceremony.
Cases in Mishna (Yebamot 2, 3-4) in which Yabam is allowed to Chalitsa and not to Ybum:
כְּלָל אָמְרוּ בַיְבָמָה. ... וְאִסּוּר קְדֻשָּׁה, חוֹלֶצֶת וְלֹא מִתְיַבֶּמֶת. אִסּוּר קְדֻשָּׁה, אַלְמָנָה לְכֹהֵן גָּדוֹל, גְּרוּשָׁה וַחֲלוּצָה לְכֹהֵן הֶדְיוֹט, מַמְזֶרֶת וּנְתִינָה לְיִשְׂרָאֵל, וּבַת יִשְׂרָאֵל לְנָתִין וּמַמְזֵר:
... a general rule has been laid down in respect of the deceased brother's wife: wherever she is prohibited... by virtue of holiness, she must perform the halizah and may not be taken in levirate marriage. ... prohibited by virtue of holiness' [refers to the following forbidden categories]: a widow to a high priest; a divorced woman, or one that had performed halizah to a common priest; a female bastard or a nethinah to an israelite; and a daughter of an israelite, to a nathin or a bastard.
The Gemara explained that both (Ybum and Chalitsa) are permitted in Torah, but here Ybum is prohibited Gzera atu Biah shenia (See Tosfot, what is biah Shenia). We see that Ybum is a Mitsvat Asse and supersedes the Lo taasse of almana leKG or mamzeret leisrael. We are coming to that later.
Can a man who cannot make Chalitsa perform a Ybum?
Mishna and Gemara in the second chapter of Sanhedrin dealt a very similar issue.
Mishna Sanhedrin 2, 2:
הַמֶּלֶךְ ... לֹא חוֹלֵץ וְלֹא חוֹלְצִין לְאִשְׁתּוֹ . לֹא מְיַבֵּם וְלֹא מְיַבְּמִין לְאִשְׁתּוֹ.
he [the king] may not perform halizah nor may it be performed to his wife. he may not perform yibbum {because Chalitsa is an humiliation and one who cannot be released by chalitsa cannot be "married" by Ybum, Bartenura and Rambam} nor may it be performed to his wife {because she can not marry again, Rashi; since she can not marry, she is not concerned by Chalitsa. The explanation of Rashi is valuable according to Rabanan, but not according to Halacha. Halacha is that she can be married by an other king (for this we agree to Rabbi Yehuda). If this other king is a Yabam, after all, since we can not make a Chalitsa between the two, since this other king needs honors, he can not make Yibbum with this widow of his brother (who was the first king), thist last explanation is on Rambam's commentary on our mishna}.
We learn from the Rambam, cited in Tosfot Yom Tov that one who is unable perform Chalitsa is also unable to perform a Ybum.
Let's examine the Rambam carefully: Sefer Shoftim, hilchot Melachim, Chapter 2, halacha 3:
וְאֵינוּ חוֹלֵץ--שֶׁנֶּאֱמָר "וְיָרְקָה בְּפָנָיו" (דברים כה,ט), וְזֶה בִּזָּיוֹן; וְאַפִלּוּ רָצָה, אֵין שׁוֹמְעִין לוֹ--שֶׁהַמֶּלֶךְ שֶׁמָּחַל עַל כְּבוֹדוֹ, אֵין כְּבוֹדוֹ מָחוּל. וְהוֹאִיל וְאֵינוּ חוֹלֵץ, אֵינוּ מְיַבֵּם; וְכֵן כֵּיוָן שְׁאֵי אִפְשָׁר לְיַבַּם אֶת אִשְׁתּוֹ--כָּךְ אֵין חוֹלְצִין לָהּ, אֵלָא תֵּשֵׁב לְעוֹלָם בְּזִיקָתָהּ.
... He may not perform chalitzah, for concerning that ritual, it is said Deuteronomy 25:9: 'And she shall spit before him.' This would be disrespectful to the king. Even if he desires to perform this mitzvah, he is not given the opportunity because a king's honor must be preserved even though he is willing to forgo it. Since he is not allowed to perform chalitzah, he is not eligible to participate in yibbum. Similarly, in the event of his death, since it is forbidden to initiate yibbum with his wife, chalitzah is also not performed for her. Rather, she must remain in her state of attachment forever.
But this principle established by Rambam (in bold characters above) is not written in Gemara.
Acharonim (including Tosfot Yom Tov) on the Mishna of Sanhedrin asked this.
The Rashash on Mishne Torah (mekorot Harambam) and on Mishnayot Sanhedrin (here) reports the same reasoning seen Rambam in Tosfot Yebamot 44a. Tosfot is discussing a Girsa. The case is: A man who deceased and left two widows without children. The Gemara (i.e. the discussed Girsa of the Gemara)asks: Why not to make Chalitsa to one wife and Yebum to the second? The Gemara answers: The verse says "who do not build", "Who has not built could not build". Tosfot says that this girsa is wrong because this case is obvious and we do not need this verse. The explanation of Tosfot is: After the Chalitsa of the first widow, the Yabam cannot make Chalitsa to the second widow (בית אחד הוא חולץ ואינו חולץ שני בתים, he makes chalitsa for one house and he doesn't make chalitsa for two houses). Since he can not make Chalitsa to the second wife, he cannot make her Ybum.
Anyway the reasoning of Tosfot shows that he agrees to the principle written in Rambam.
In conclusion, Tosfot and Rambam read this material implication as a logical equivalence.
With Isur Keddusha, Yabam makes Chalitsa but not Ybum.
In Gemara Yavamot 20a, there is an opinion, apparently rejected at the end of the Sugia, the opinion of Rav Gidel in name of Rav. He rules from the verse "Yebimto" that Isur Kedusha, that is Chayve Lavin have Chalitsa and not Ybum. An exception against the Rule "כל העולה ליבום עולה לחליצה, כל שאינו עולה ליבום עולה לחליצה" (A Yabam).
R`Giddal replied in the name of Rab: Scripture stated, Then his brother's wife shall go up to the 'gate, where there was no need to state his brother's wife; why then was 'his brother's wife' specified? [To indicate that] there is a case of another brother's wife who goes up for halizah but does not go up for levirate marriage.(46) And who is she? One of those prohibited by a negative precept.
But the conclusion of the Gemara is that Miderabanan the cannot make Ybum wiiht Isur Keddusha. but min Hatora it's permitted. (see Rambam Ybum 6.10).
Difference between Issur Kedusha (lav) as Mamzeret and King:
After all the Yabam can make Chalitsa to a Yebama, so why a king cannot make a Ybum? In the case of Isur Kedusha both Y. and Ch. are regulatory. In the case of a King, the action of Chalitsa is itself a sin, despite that it is a Mitsva. so Chalitsa can not reject its proper problem, no Ase Doche Lo Taase. This is the reasoning of Chachamim against Rabbi Yehuda. Since Chalitsa and Ybum are prohibited Min Hatora.
See shut Ber Ytschak from Rav Ytschak Elchanan Spector Mikovno
Conclusion:
Case: The Yabam has no legs and Chalitsa is not possible. The woman will remain Aguna. I think that such Yabam is not allowed to make Ybum at all (But may be that my demonstration is wrong d/t difference between technical impossibility and halachical prohibition.)
There is an other possibility to understand the Rambam. In Gemara we have two statements.
(1) One who can make ybum can make chalitsa:
A => B
(2) One who cannot make ybum cannot make chalitsa
~A => ~B
So, a logical result of the both statements is
A <=> B
~B <=> ~A
A man man who is not able to perform chalitsa is not able to perform ybum.
