Can cousins get a green card when they get married in America?

You can get a green card in this case.

Any immediate family member of an American citizen can get a green card at any time. As long as he marries an American, there is no restriction on getting a green card.

However, the US Immigration Bureau attaches great importance to the authenticity of marriage, and applicants need to provide a lot of evidence to prove the authenticity of marriage, such as financial documents, marriage certificates, joint property or lease contracts, insurance policies, bills, etc.