The lectionary readings of the EF Mass:

Can the priest read/chant the Epistle and the Gospel in the vernacular as he gets to them during the course of the Mass or must he read/chant them in Latin?

According to the Instruction Universae Ecclesiae (2011), at a Low Mass, the priest may read the readings in the vernacular.  However, most often the readings are read in Latin and repeated in the vernacular before the homily.