Saint James School of Medicine seems legit. It is a private, for-profit medical school. They appear on Wikipedia's list of medical schools in the Caribbean, and they are ECFMG eligible, meaning that their graduates may be considered for residencies in the US (and, it seems, Canada as well). The letter also seems legit, in that they are directing you to send payment to the school, not to some random Venmo account.
The catch is that actually getting a residency in the US can be quite difficult. At reputable schools in the US, it is typical for 95% of students to match. At schools like St. James, the number is closer to 65%.
I notice also that they claim 89% of their students passed the USMLE (licensing examination) on the first try, but Wikipedia reports that the real number is closer to 35%. The discrepancy is explained by the fine print -- the 89% only counts students who got 195 or higher on the Comprehensive Basic Science Exam. If your cousin has never taken science "in his life," then it's unlikely he will be in this group.
In short: it looks like your cousin has a pathway toward earning a "real" MD. But, becoming a board-certified, licensed doctor requires more than an MD, and that might be a real challenge. This could lead to a difficult situation where your cousin has spent a fortune on tuition and fees but cannot see much return on that investment.