This sort of thing should work:
CREATE PROCEDURE pr_Prize_Winner
AS
SET NOCOUNT ON;

DECLARE @User_Id INT = (
         SELECT TOP 1
            u.User_ID
         FROM
            [User] u
         JOIN  Issue i ON u.User_ID = i.Reporter_ID
         WHERE
            i.Date_Submitted BETWEEN DATEADD(DAY, -7, GETDATE()) AND GETDATE()
            AND u.HRD_Staff = '0'
         ORDER BY NEWID()
        );
INSERT TargetTable(User_Id, First_Name, Last_Name)
SELECT User_Id, First_Name, Last_Name
FROM ...
WHERE User_Id = @User_Id

Notes
1) It is best practice to schema-qualify your table and proc names.
2) Using reserved words as column names (User_Id) is not recommended
3) You may want to issue a command to TRUNCATE the target table before running the INSERT.