Do Catholic priests with the faculty of performing marriage also get to perform annulments, or does that need referral to the bishop?
An annulment, strictly speaking, is not "performed", and this is a very common misuse of terminology. There is not a marriage that is made null, but a judgment is issued that the relationship in question was never a marriage. This judgment is typically called "a writ of nullity".
There is a quite complicated procedure to go through to obtain a writ of nullity for a marriage. This procedure is dealt with primarily in canons 1671–1707 of the Code of Canon Law. In 2015, Pope Francis, who as pope is the Supreme Legislator of the Church, issued a letter titled "Mitis Iudex" ("The Gentle Judge") updating some of these canons in an effort to make it easier to obtain a decree of nullity, particularly when the nullity of the relationship is obvious or where the load of cases is heavy. What it amounts to is this:
In keeping with Vatican II's emphasis on the bishop as the primary shepherd of the diocese, the bishop is recognized as the primary judge of such cases. (Canon 1673 section 1, as amended by "Mitis Iudex")
He may exercise his power either on his own, or through a panel of judges (Canon 1673 section 3)
If both spouses agree that the relationship is not a valid marriage, and there are circumstances which make this obvious, the bishop has power to execute a briefer annulment process and make the declaration on his own (Canons 1683–87)
Otherwise, the case is referred to a tribunal of three judges, led by a member of the clergy (Canon 1673 section 3). This tribunal is constituted by the judicial vicar, a legal official appointed by the bishop. This tribunal will be the agency that hands down a writ of nullity.
Before this letter was issued, a judgment by a tribunal that the relationship was not a marriage would be automatically appealed to a second tribunal. This made the process significantly longer, without necessarily changing the final judgment. An appeal from the first tribunal's decision can still be made, but is no longer made automatically. The appeal will be made to the metropolitan archbishop, unless the decision was made by the tribunal of the metropolitan archdiocese, in which case the appeal will be to the Roman Rota. In theory, of course, a final appeal could be made to the Pope.
All canon law citations refer to the updated canon law found in the body of "Mitis Iudex". Similar updates were made to the Code of Canons of the Eastern Churches by a letter issued at the same time as "Mitis Iudex" and titled "Mitis et misericors Iesus".
No, not even a bishop can perform annulments.
Canon 12. If anyone says that matrimonial causes do not belong to ecclesiastical judges, let him be anathema.
Thus, positively, judging whether a marriage is valid or not belongs to the Church (specifically, to marriage tribunals; cf. What God Has Joined Together: The Annulment Crisis in American Catholicism by Robert H. Vasoli).