Can someone blow a Shofar during the Omer at times when playing instruments is prohibited (i.e. after Passover and before Lag b'Omer, according the predominant custom)?
Even though a Shofar is considered a musical instrument, and thus is forbidden to blow in Shabes (not allowed from Derabanan to prevent the situation where a person will fix the intrument), but there is no ground to deny blowing during Sfiras HaOmer sihce it is not "an act of joy" and like sad melodies are not forbidden in the three Bein ha-Metzarim weeks, so is the Shofar blowing not forbidden then, all the more so in Sfiras HaOmer.
Based on the answer to the question Live Vs. Recorded Music during the Omer it seems that simply practicing to blow shofar would be permitted.
If you plan on making lively music with a shofar then the Halacha may be more stringent, as the basis for no-music during the Omer comes from the Kitzur Shulchan Aruch and Aruch Hashulchan that forbids dancing even at engagement parties:
אַךְ רִקּוּדִין וּמְחוֹלוֹת, אֲסוּרִין