Can a stolen shofar be used to fulfill the mitzvah on Rosh Hashana?
Obviously a stolen shofer should not be used on Rosh Hashana, based on the established Talmudic dictum, אין מצוה הבאה בעבירה, i.e. we do not carry out a mitzvah if doing so would involve committing a transgression. However, if someone did indeed steal a shofar and use it to fulfill the mitzvah, he does fulfill the obligation.
The reason is because, technically speaking, the mitzvah is actually not to blow the shofar, but to hear the sound of the shofar blown. What we really need is the sound produced by the shofar, not the shofar itself, and there is no such thing as stealing sound.
Can you use a borrowed shofar or do you have to be the owner, i.e. do you have to buy your own shofar? To do the mitzvah of taking hold of the Four Species on Sukkot, you must be the owner of the Four Species, since the verse which prescribes the mitzvah includes the word לכם (Vayikra 23:40), which comes to teach us that the Four Species must actually belong to you.
Likewise regarding shofar we find the word לכם (Bamidbar 29:1). However, regarding the mitzvah of shofar, the verse is a bit different.
יום תרועה יהיה לכם
Here we are told that there is to be a day of תרועה, of shofar blasts. Had it said ולקחתם לכם שופר — and take for yourselves a shofar, like the Torah instructs us to take hold of the Four Species on Sukkot — then we would understand that the shofar must belong to us. Since the shofar does not have to be yours, even if you do not buy a shofar, but borrow it without permission you can fulfill the mitzvah, since we have a rule that it is to one’s advantage to allow another person to perform a mitzvah with his possessions.