When the Shaliach Tzibbur says Tachanun, should he leave his position and say it sitting down or should he rather stay in position and say it without physically sitting down but in a manner which is still halachically valid for Tachanun? Related: Tachanun with No Seats
Per Rabbi Naftali Hoffner in Halacha - Nesias Kapayim U'Tachanun 92:3 the three steps one is supposed to take after Shemona Esrei prior to walking is applicable to the Chazan. The Chazan does these three steps at Kadish Shaleim after the end of Ashrei / Uva Ltzion. Therefore it is proper for the Chazan to not sit down and to say Tachanun at the Amud. He does note that according to the Rambam the Chazan may sit down for the Tachanun.
Tachanun is the only part of davening where the chazzan does not stand. Since the entire purpose of the Tachanun is to recite a prayer while one is bowing, the chazzan also "falls Tachanun."
This answer states that if one is leaning on the amud to the point where he would fall if it were removed, then, it's considered like sitting. Thus, I have seen many shlichei tzibbur do this.
As to whether it is ideal for one to sit fully on a chair and therefore leave the amud to do this - I'm uncertain. I've seen both methods performed. One may question if leaving the amud for something possibly unnecessary such as this might be torach tzibbur.
Apparently, he doesn't need to leave his position, as I've alluded to, but, is better clarified in the linked question. As to whether it is ideal for the chazzan to specifically sit, and if he needs to, can he leave the amud to do that - I don't know, offhand. I have seen chazzanim do that, while others lean, and, both are permitted.