The visually impaired person groped their way toward the exit.
Literal
Eye [poss-の] not-free that person [topic-は] exit [poss-の] direction [toward-へ] groping [by-で] went.
手探りで ('by groping, feeling with one's hands') uses で to mark the method of movement. ~の方へ ('toward the direction of') is a softer directional than plain へ alone.