Not sure what you think home schooling has to do with earned income credit. Why are you asking about that?
https://www.irs.gov/credits-deductions/individuals/earned-income-tax-credit/use-the-eitc-assistant
EDUCATION EXPENSES
Education expenses for grades K-12 are not deductible on your Federal return, whether for private, public, or home schooling, nor are the expenses for tutoring, after school lessons or after school activities, such as dance lessons, sports, etc. Some states allow deductions/credits for K-12 education and/or home schooling expenses; if your state has these deductions available, you will be prompted to enter them when you prepare your state return. (As far as I know, the states that offer any sort of K-12 deductions/credits are Arizona, Illinois, Indiana, Iowa,Louisiana, Minnesota, and Wisconsin)
To meet IRS requirements of a "school". If in school full time 5 months parent may be eligible for EIC. If "Home School" "High School" does not meet IRS designation of a school like they list than it is not eligible. But the rules do not specifically state home school does not meet the eligibility requirements of a "school". I called support and have no answer either. IRS does not answer phone at this time. If someone has contacted IRS regarding this I would greatly appreciate it!
How old is your child? If they were under 19 on Dec 31 it doesn't matter.
Turned 19 in summer after graduating Home High School so does matter for us. Need to know if IRS accepts a High School Home School as a school.
There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit.
A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:
Q. Does IRS accepts a High School Home School as a school, for purposes of the QC (and EIC) full time student rule?
A. No.
Student defined. To qualify as a student, your
child must be, during some part of each of any 5
calendar months of the year:
1. A full-time student at a school that has a
regular teaching staff, course of study, and
a regularly enrolled student body at the
school; or
2. A student taking a full-time, on-farm training course given by a school described in
(1), or by a state,
Full-time student. A full-time student is a
student who is enrolled for the number of hours
or courses the school considers to be full-time
attendance.
School defined. A school can be an elementary school; junior or senior high school;
college; university; or technical, trade, or mechanical school. However, an on-the-job training course, correspondence school, or school
offering courses only through the Internet
doesn’t count as a school.
Vocational high school students. Students who work on “co-op” jobs in private industry as a part of a school's regular course of
classroom and practical training are considered
full-time students.
Reference: Publication 17; pages 28-29
https://www.irs.gov/pub/irs-pdf/p17.pdf
Turned 19 in summer after graduating Home High School
Take a look at IRS Publication 17 page 27 at https://www.irs.gov/pub/irs-pdf/p17.pdf. In the table under "Tests to be a qualifying child" item #2 states "The child must be (a) under age 19 at the end of the year and younger than you (or your spouse, if filing jointly); (b) under age 24 at the end of the year, a student, and younger than you (or your spouse, if filing jointly); or (c) any age if permanently and totally disabled."
If your child was a student for at least 5 months in 2021, this requirement is met. If they "graduated" on May 1st, then the entire month of May counts.
As for the EIC, since your child turned 19 in 2021, they don't qualify you for the EIC. They have to be "under" the age of 19 at the end of the tax year in order to qualify. See https://www.irs.gov/credits-deductions/individuals/earned-income-tax-credit/qualifying-child-rules#Tests%20for%20a%20Qualifying%20Child