The original sentence has a subject-verb agreement issue. "Visitors and groups of students' coming every year bring them a lot of money and a better life." Here, the subject "Visitors and groups of students" is plural, so the correct verb form should be "bring" instead of "brings."
Your corrected sentence, "Visitors' and groups of students' coming every year brings them a lot of money and a better life," still has a possessive issue. Using "Visitors' and groups of students'" suggests possession, but it seems like you want to express the action of coming belonging to both visitors and groups of students. In this case, you don't need the possessive apostrophe.
A corrected version could be: "Visitors and groups of students coming every year bring them a lot of money and a better life."