Do OBGYNs have to perform abortions?

No. There are often conscience clauses that allow people to not perform surgeries that are against their will.

However all ob/gyns should get training on abortions in case they need to help terminate an incomplete abortion - a miscarriage gone wrong to save the life/uterus of a woman. But NOT all ob/gyn programs require that training or provide it.

Medical Students for Choice might have more information on which residencies require abortion training.