All instances of "they" and "their" refer to the people who live on earth, all instances of "him" and "he" refer to God, and the word "us" includes the speaker, the hearers, and the people who live on earth. (See: [[rc://en/ta/man/jit/figs-exclusive]])
Here "search for God" represents desiring to know him, and "feel their way toward him and find him" represents praying and having a relationship with him. Alternate translation: "so that they should want to know God and perhaps pray to him and become one of his people" (See: [[rc://en/ta/man/jit/figs-metaphor]])