The term "evangelist" is an Anglicized form of the Greek euanglistes, derived from eu ("well") and angelos ("messenger").
An evangelist is someone who proclaims the good news, serving as a gospel herald. In a broader sense, all Christians are evangelists.
However, the term is used in the New Testament in a more restricted sense—equivalent to a gospel minister (cf. Acts 21:8; Eph. 4:11; 2 Tim. 4:5).
In this latter sense, a woman cannot serve as an evangelist (1 Cor. 14:34; 2 Tim. 2:12). In the general sense, she can and must.