Chazan of Musaf: The Chazan of Musaf is not to blow the Shofar during Musaf as perhaps he will become flustered and will be unable to continue his prayers without confusion. However if he is positive that he will not become confused in his prayer and will be able to continue properly then it is allowed for him to blow the Shofar. Likewise if the Chazan is praying from a Siddur then he may blow the Shofar as he will certainly be able to continue his prayers without confusion. In such a case he may be the Baal Tokeia even if there are others available that know to blow as well as him.
Person who blew after Shacharis: There is no longer an obligation today to honor the Baal Tokeia of Meyushav with the blowing of Musaf, and rather another person may be given the honor to blow. [However some Poskim rule that those areas which have a custom to honor the blower of the first set with the blowing of the second set blown during Musaf must honor this custom.]
Makri for Musaf:
It is not the Chabad custom to have a Makri point at the sounds by the blows in Musaf [both in the silent and repetition] or after Musaf.
General Q&A on the Musaf blows
Does one have to listen to the entire Chazarah of the Chazan or may he learn Torah or recite Tehillim?
One is to listen to the parts of the repetition that are obligatory and is not to do anything else in the interim. Regarding the Piyutim and verses of Malchiyos, Zichronos and Shofros, it is forbidden for one to learn Torah verbally during this time. This applies even prior to Chazaras Hashatz. One may however think words of any part of Torah without limit. One may even verbally learn matters of Torah that relate to the Tekios. Some also permit one to say Tehillim during this time.
 592/10; Michaber 585/4
 Admur ibid; M”A 585/9; M”B 585/14
 M”A 585/10; Birkeiy Yosef 585/8; Kisei Eliyahu 585/4; Erech Hashulchan 585/5; M”E 585/11; M”B 585/15
The reason: The reason it is permitted for the Chazan to blow the Shofar and is not considered an interval in his prayers is because in truth the blows are considered part of the prayers being that he is required to blow in the order of the blessings. It is thus not considered an interval at all within his prayers. [Admur ibid; M”A 585/8 as explained in Machatzis Hashekel ibid; Chayeh Adam 141/10; M”B 585/13]
Other opinions: Some Poskim rule that if there are others available who can blow then the Chazan may not blow the Shofar even if he is certain that he will be able to return to his prayers without confusion. [Bach in name of Rashal; Peri Chadash brought in M”A 585/10; Kaf Hachaim 585/42]
The law if he will become flustered but there is no one else who knows how to blow: Some Poskim rule in such a case he is to blow and Hashem will help that he return to his prayer. [Taz 585/5; Shivleiy Haleket; Hatanya; Elya Raba 585/14] Others rule he is not to blow during Musaf and is to delay it until after Davening. [M”A 585/9]
 592/10-11; Michaber 592/4; Rama 585/4; See Otzer Minhagei Chabad 327
In previous generations it was accustomed to give the honor of blowing to a single person and hence this person merited all the Tekios [of that day] even those that are blown by the blessings of Musaf and one should not remove the Mitzvah from him. [592/11; Michaber ibid] The reason [for why they gave it to a single person] is because one who begins a Mitzvah is told to complete it and hence it remains proper that the person who blew the Tekios of Meyushav should also blow the Tekios of Meumad, unless he is the Chazan of Musaf and is not positive that he can continue his prayer without confusion. [592/10; Rama ibid] However in today’s times that some places give the Tekios of Meumad to another person it comes out that he never merited all the Tekios and it is hence permitted to be given to another person in all areas. [592/11; M”A 592/7; M”B 592/17] Furthermore even in previous times this matter was only initially required however Bedieved if the Tekios were delegated to someone else the obligation is fulfilled. [Rama ibid; Shulchan Gavoa 592/11] See Taz 585/6; Mateh Efraim 585/12 that some are accustomed to distribute the Tekios of the three different blessings to three people.
 The above wording in Admur ibid and M”A ibid imply that there is no need to even heed by the dictum “One who begins a Mitzvah is told to complete it”. Vetzaruch Iyun as although he may not merit the Tekios why shouldn’t it remain proper to have him do it in order to fulfill this dictum of the Sages.
 Mateh Efraim 585/12; 592/8
 Otzer Minhagei Chabad 328
Other customs: The widespread custom is to have a Makri by the blows sounded in the repetition of Musaf and after Musaf. [See M”E 592/9 and 11; Mishmeres Shalom 41/7; Likkutei Mahrich 70b] One may certainly however not have a Makri recite the words during the silent prayer of Musaf. [Minchas Elazar 4/37; Sdei Chemed R”H 2/16] However some are accustomed to be lenient to have a Makri even during the silent Shemoneh Esrei. [See Otzer Minhagei Chabad 328; Piskeiy Teshuvos 585 footnote 68]
May the Chazan of Musaf be the Makri? Some Poskim rule that the Chazan of Musaf may not be the Makri for the Tekios due to it being an interval. [Shaareiy Tziyon 585/30 in name of Peri Megadim] However some Poskim argue that even the Chazan may be the Makri and it is not considered an interval. [Levushei Mordechai 4/37; See Elya Raba 585/15]
 Admur 124/6; Minchas Yitzchak 4/48; See ibid and Piskeiy Teshuvos 592/1 regarding the novelty of the Griz of Brisk that one must listen to the parts that invalidate the prayer if they are omitted in order to be Yotzei the Tekios of Musaf. However the Chazon Ish 137/3-5 negates this novelty. See Elya Raba 592/8 that one may walk around during the repetition so long as he does not talk or remove his mind from the prayer, however some are even stringent not to move from their place. [brought in Kaf Hachaim 592/26]
 Har Tzevi 1/42; Piskeiy Teshuvos 592/4; See Minchas Yitzchak ibid and 4/47; as one may not make an interval between the sets of Tekios.
 Minchas Yitzchak 4/48 in explanation of the Leket Yosher from the student of the Terumas Hadeshen p. 128
 Yesod Veshoresh Havoda 11; Piskeiy Teshuvos ibid; however perhaps there he is only referring to Chazaras Hashatz of Shacharis, Vetzaruch Iyun; See however Otzer Minhagei Chabad p. 128 that the Rebbe at times would say Tehillim during the Chazan’s recital of Hineni Heani. Seemingly the reason this is not considered an interval is because it is similar to matters relating to prayer which one may even initially speak of during this time.