The verb should be "bring" to match the plural subject "groups", so the second one isn't grammatically correct. They both have punctuation errors. There shouldn't be apostrophes after either "visitors" or "students."

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."

You can use singular or plural. They are both correct. The more conventional way to do it would be to use the plural since the subject of the sentence is a group of many people. However, it is permissible to conceptually regroup a plural into a singular entity. Here's a better example: "Alice baked the best cookies. Letting the dough sit for a long time and not adding too much sugar was the secret that made them so good." Technically, you should say "were the secrets" but it is natural to group these two secrets into one singular secret. Omit the apostrophes.

no ' needed bring