You seem to be describing a cundina, described as:
A small group, anywhere from 5-12 people, will agree to make monthly contributions into a common fund and will proceed to take turns borrowing from that fund, one person at a time. Members do not charge each other interest, as their main motivation to participate is reciprocity.
This description mentions borrowing, but most cundinas are set up so that participants simply take turns getting the whole sum each month.
The word cundina is originally Spanish, but it is being used more and more in the U.S. to describe this kind of micro-lending structure. That said, the word is not yet widely known among English speakers.
See here for more.
Other cultures have their own words for this.