Are the following sentences correct?
He's a genius! He got a scholarship to study at Oxford.
Very few students can be scholars at Yale University.
Your first sentence is fine. You could also simply say "a scholarship to Oxford", since "to study" is implicit.
Your second sentence is also fine, but is ambiguous; it can mean:
1. Very few people can win a scholarship to Yale.
2. Very few people, once they win a scholarship, can successfully complete their studies at Yale.
3. I would guess that there are very few scholars at Yale.
(Some of these statements are more controversial than others.)
All the best,
Not a professional ESL teacher.