No, Catholic priests are not the only ones who can do exorcisms. The authority to cast out demons and evil spirits rests not in a church structure but in the authority given to Christians who are redeemed by the blood of Christ. Christians have the authority to be called the children of God (John 1:12).
Jesus gave the authority to the disciples over unclean spirits (Matthew 10:1; 18:18) because he had all authority in heaven and earth (Matt. 28:18). Jesus cast out demons (Matt.12:27; Mark 1:23-26, 5:1-13; Mark 9:25-26; 16:17; Luke 9:1; 10:17; 11:19), as did Paul (Acts 16:16-18) and the Jews (Acts 19:13).
Since Christians have the authority to be the children of God, we also have the right to call upon the name of the Lord Jesus Christ, and with the authority that he possesses Christians have the right to be able to perform exorcisms -- though such an endeavor should never be taken lightly.