The Gemoro in Rosh Hashono 17b says that HKB”H put on a talis like a shaliach tzibbur and taught Moshe the 13 middos and said whenever Israel sins they should perform before Me according to this order (which we understand to be repeating the 13 middos) and I will forgive them. Are there any authorities who infer that for saying the 13 middos the shaliach tzibbur should wear a talis?
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Yes, Taame Haminhagim 681 says (with my own translation):
(However, the footnote there cites other reasons the shatz should do so, and other views as to whether he should.)
The minhag at Yeshivat Har Etzion is for the Ba'al Keriyah to cover his head with his tallit when reading the aliyah that contains the 13 middot (say, on a fast day).