I currently am enrolled in Penn Foster Veterinary Technician program. I pay monthly out of pocket for the tuition, does this count as student loan payments?

No.    If you have a student loan and are making loan payments, you get a 1098E for the interest you paid when you made payments for the loan.   It sounds like you are paying tuition out of pocket on a monthly basis.

To see if a school is eligible to take education credits

Enter your school at the link below, to see if it's on the dept. of education list.

https://ope.ed.gov/dapip/#/home

https://ttlc.intuit.com/questions/1901172-what-are-education-tax-credits

https://ttlc.intuit.com/questions/1901172-what-education-tax-credits-are-available      

https://www.irs.gov/help/ita/am-i-eligible-to-claim-an-education-credit

https://ttlc.intuit.com/questions/3262983-who-is-eligible-to-take-the-american-opportunity-tax-credit

AOC will be shown on line 29 of your Form 1040

https://ttlc.intuit.com/questions/3383321-why-didn-t-i-get-a-credit-or-deduction-for-education-expenses

And....you have not mentioned your age or whether you are being claimed as a dependent on a parent's tax return.  If you can be claimed as someone else's dependent, they are the ones who can use the education credit on their tax return.

Q. I pay monthly out of pocket for the tuition, does this count as student loan payments?

A. No. But, it counts as paying tuition for purposes of claiming a tuition tax credit.