Educational Opportunity and Success Act of 2021 This bill reauthorizes through FY2027 and otherwise revises TRIO programs. (These outreach and student-services programs identify and provide services to students from disadvantaged backgrounds.) Among other revisions to the programs, the bill prohibits the Department of Education (ED) from rejecting TRIO grant applications based on certain errors; requires ED to provide additional technical assistance to interested grant applicants; revises the outcome criteria for measuring the quality and effectiveness of the programs, including those programs specifically designed for veterans; allows program administrators to use a student's most recent Free Application for Federal Student Aid to determine TRIO program eligibility; and increases the maximum stipend for students participating in the Upward Bound Program or the Ronald E. McNair Postbaccalaureate Achievement Program.