Are you allowed to shop/buy sale items on 17th of Tammuz fast?
It is permitted, especially if they are items that are not very expensive and will not being the person a lot of joy.
See Biur Halacha 551-2 D:H M’rosh Chodesh that he writes from the Eliyahu Rabba that it is possible that one should be careful on !7 B’Tammuz, and 10 B’teves like the halachos of the nine days (when shopping should not be done). However see Daas Torah 551-2 that what the Biur Halacha is bringing is based on a printing mistake, and also see Shoneh Halachos 551-38, that Horav C. Kanievsky also brings up this point. According to them, there wouldn’t even be an inyan to be machmir. Also see Chut Shani Shabbos 4- pg. 381, and Kara Olai Moed chap. 1 ftnt. 15.
Aside from this, it is sceptical that buying sale items is included in what can’t be done during the nine days. See M:B 551-11 that it is a machlokes if masa umatan that is ossur in the nine days refers to business of simcha or all business. In addition, he also says that if it won’t be bought now, such as an item that is on sale then everyone agrees that it is permitted. Therefore with all of this we don’t have to be machmir, when even the B:H only considered it questionable.