The study on minimal linear codes has received great attention due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, to the best of our knowledge, no infinite family of minimal ternary linear codes was found from vectorial functions. In this paper, we present a necessary and sufficient condition for a large class of ternary linear codes from vectorial functions such that those codes are minimal. Based on that, we construct several minimal ternary linear codes with three-weight from vectorial regular plateaued functions, and determine their weight distributions. Moreover, we also give a necessary and sufficient condition for a large family of ternary linear codes from vectorial functions such that the codes are minimal and violate the AB condition simultaneously. According to this characterization, we find several minimal ternary linear codes violating the AB condition. Notably, our results show that our method can be applied to solve a problem on minimal linear codes proposed by Li et al.