I don't know if this culture exists in English speaking countries, but in my country a group of people would contribute a set sum of money every month so that each one of the group receives the total sum of the group's contributions. This is some kind of a lending system to which people resort so they can use the money collected to get married or buy a car or similar things. Is there a word for this?
This sounds like pooling to me.
Sometimes people can pool their resources. M-W defines the word as:
pool (v.) to combine (as resources) in a common pool or effort
whereas Cambridge defines this as:
pool (n.) a number of people or a quantity of a particular thing, such as money, collected together for shared use by several people or organizations:
Everybody puts some money into a common pool.
We need a reserve pool of cash, just to be on the safe side.
Note: In many English-speaking countries, this usage of pool often refers to a gambling pool, where people put money into a "pot" that the winner collects; such pools often are related to sporting events. However, the word could also be used in the way you describe:
Let's all pool our money together so that there will be a fund to draw from when one of us has to plan a wedding or a funeral.
If I understand you correctly, this form of financial scheme is known as:
rotating savings and credit association.
A rotating savings and credit association (ROSCA) is a group of individuals who agree to meet for a defined period in order to save and borrow together, a form of combined peer-to-peer banking and peer-to-peer lending.
F.J.A. Bouman described ROSCAs as "the poor man's bank, where money is not idle for long but changes hands rapidly, satisfying both consumption and production needs." They are also known as tandas (Latin America), partnerhand (West Indies), cundinas (Mexico), 'Hagbad (Somaliland)'susu (West Africa and the Caribbean), hui (Asia), Game'ya (Middle East), kye (계) (South Korea), tanomosiko (頼母子講) (Japan), pandeiros (Brazil), juntas (Peru), or quiniela.
The most common term for this sort of thing is savings pool or money pool. Examples:
In India, I have heard this called a 'chit fund' and it is considered an example of a 'cooperative lending society'.
From Wikipedia quoting Indian legislation:
A transaction ... under which a person enters into an agreement with a specified number of persons that every one of them shall subscribe a certain sum of money (or a certain quantity of grain instead) by way of periodical installments over a definite period and that each such subscriber shall, in his turn, ... be entitled to the prize amount.
From the Oxford online dictionary, close to the above definition:
An institution which accepts savings at interest and lends money for house and other purchases
In South Africa, we call them Stokvels. Technically not English, but it has become a common word for English speakers in SA.
I think you're talking about a susu https://en.wikipedia.org/wiki/Susu_(informal_loan_club), and although I couldn't find that word in any English dictionaries, I can confirm that some people do just use that word in English, and some English speakers know what it means.