Can you buy me a cup of coffee?
Will you buy me a cup of coffee?

Which of the two is the correct way of making a request?

Well, there are many ways of making requests.

Can you and Will you are less polite ways of asking someone to do something and are used in informal/casual situations.
Can you pass me the pepper?
Will you pass me the pepper?
However, to make your requests more polite in a casual way, you can add please to the statement.
Can you please buy me a cup of coffee?
Will you pass me the pepper, please?

Could you and would you are more polite ways of asking someone to do something and are used more in formal situations.
The boss says:
Could you open the windows?
Would you send me that file at the earliest?
( please can always be added to increase the degree of politeness)

Could is also used to make polite requests in informal situations.
Could you buy me cup of coffee?

To sum up,
In terms of increasing degree of politeness:
will < can < could< would
In terms of increasing degree of formality of situation:
will/can < could < would