The Rambam writes that a stolen shofar is not disqualified because of mitzvah haba'ah b'aveira because "ain b'kol din gezel," the mitzvah object required for the mitzvah of tekiyas shofar is sound, and sound is not an "object" which can be stolen. Ra'avad writes that the source for the Rambam's din is not a sevara, but rather a derasha (quoted in the Yerushalmi), "Yom teru'ah yehiyeh lachem," even if stolen.
Last week we learned from the Mishnas Ya'avetz that there are two dinim in the psul of mitzvah haba'ah b'aveira: 1) failure of the individual to fulfill his mitzvah, a psul gavra; 2) a disqualification of the object from use in mitzvah performance.
Based on this we can understand why the Rambam cites a sevara in addition to the derasha of the Yerushalmi. The Yerushalmi teaches us that a stolen shofar is not disqualified as a mitzvah object -- it is not like a cracked shofar or a shofar that is too short. However, that does not necessarily mean a person can attain a kiyum mitzvah using such an object. The individual performing the mitzvah has attained his goal through improper means and deserves no credit. The Rambam therefore adds that the kiyum mitzvah of shofar is through the sound of the shofar only. The act of stealing the shofar object is only indirectly related to the actual mitzvah performance and does not disqualify it.