We consider a problem of enumerating chemical graphs from given constraints concerning their structures, which has an important application to a novel method for the inverse QSAR/QSPR recently proposed. In this paper, the structure of a chemical graph is specified by a feature vector each of whose entries represents the frequency of a prescribed path. We call a graph a 2-augmented tree if it is obtained from a tree (an acyclic graph) by adding edges between two pairs of nonadjacent vertices. Given a set of feature vectors as the interval between upper and lower bounds of feature vectors, we design an efficient algorithm for enumerating chemical 2-augmented trees that satisfy the path frequency specified by some feature vector in the set. We implemented the proposed algorithm and conducted some computational experiments.