Dear Rabbi Maroof,
May the Shofar be sounded on any days between Rosh Hashana and Yom Kippur?
If not, where is that written and what is the reason?
Sounding the Shofar between Rosh Hashana and Yom Kippur is not halakhically problematic. The only time it is prohibited is on the day before Rosh Hashana. This is in order to make a clear distinction between the blowing of the Shofar on the holiday, which is done in fulfillment of a Biblical obligation, and the blowing of the Shofar practiced during the preceding month of Elul, which is only a widespread custom.
It is also prohibited to blow the Shofar on Shabbat because this would constitute playing a musical instrument which is not allowed then.