Answer: The Shulchan Aruch (OC 493:1) writes that because the students of R’ Akiva died during the omer period, one shouldn’t get married or have haircuts during this time.
R’ Ovadia Yosef (Yabia Omer 3:26; Yechave Daas 1:24) writes that many people confuse this time period with the three weeks leading up to Tisha B’Av. Unlike the three weeks which is a time of mourning for the churban, the omer is primarily a time of growth leading up to Shavuos. Thus, while the Shulchan Aruch (OC 551:17) writes that one should ideally not say shehecheyanu during the three weeks on new clothes or a new fruit, the Mishna Berura (493:2) writes that one may say shehecheyanu during the omer.
R’ Eliezer Waldenberg (Tzitz Eliezer 18:41) notes that there were some communities that had the minhag to avoid saying shehecheyanu, though even they would allow it on Shabbos. Unless, one has such a minhag, however, one may buy new clothes and say shehecheyanu.
R’ Benzion Abba Shaul (Ohr Letzion 3:17:2) writes that as the Shulchan Aruch only forbade haircuts and getting married during this time, there is no prohibition against buying new clothes or big furniture (See Emes L’yaakov OC 493:n486; Halichos Shlomo, Sefiras Haomer 11:n53)
Nonetheless, R’ Ovadia Yosef (ibid.) and other poskim write that if one doesn’t need new clothes, they should wait to wear them. Thus, R’ Betzalel Stern (Betzel Hachachma 4:48) writes that one observing the first half of the omer may buy clothes in the second half and vice versa.
The Mishna Berura (551:45) writes that one can buy clothes during the three weeks that one wouldn’t normally say shehecheyanu on. Thus, there would certainly be no issue with buying new shirts, etc. during the omer.
The Piskei Teshuvos (493:3) quotes various poskim who write that one can buy a new suit if their old one needed replacing. Likewise, one may buy children’s clothes. R’ Ephraim Greenblatt (Rivevos Ephraim 8:227) adds that one may buy clothes during the omer, especially if they are on sale.
In conclusion, one may buy new clothes during the omer, unless one has a specific minhag not to. If it is something that one should say shehecheyanu upon wearing for the first time, they should ideally do so on Shabbos.