Can I claim the lifetime learning credit if I did not receive a 1040 or 8863 form?

Nobody receives a 1040 or 8863 form.  those are the forms TurboTax fills out for you to claim the credit.  You do need a form 1098-T from your school, or qualify for an exception

 “However, a taxpayer may claim one of these education
benefits if the student doesn’t receive a Form 1098-T because
the student’s educational institution isn’t required to send a
Form 1098-T to the student under existing rules (for example, if
the student is a nonresident alien, has qualified education
expenses paid entirely with scholarships, or has qualified
education expenses paid under a formal billing arrangement). If a
student’s educational institution isn’t required to provide a Form
1098-T to the student, a taxpayer may claim one of these
education benefits without a Form 1098-T if the taxpayer
otherwise qualifies, can demonstrate that the taxpayer (or a
dependent) was enrolled at an eligible educational institution,
and can substantiate the payment of qualified tuition and related
expenses.”  https://www.irs.gov/pub/irs-pdf/f8917.pdf

. 

Form 1040 and 8863 are forms that the taxpayer completes and files with the IRS.  These are not forms that you would receive from some.   To claim the Lifetime Learning Credit you (or your dependent) must be (or have been) a student who is (or was) enrolled in at least one course during the tax year, AND have a modified adjusted gross income below the threshold (for 2020, the threshold is $69,000 or $139,000 for joint filers).  If you meet these qualifications then you can claim a lifetime learning credit.

Form 1040 and 8863 are forms that the taxpayer completes and files with the IRS.  These are not forms that you would receive from some.   To claim the Lifetime Learning Credit you (or your dependent) must be (or have been) a student who is (or was) enrolled in at least one course during the tax year, AND have a modified adjusted gross income below the threshold (for 2020, the threshold is $69,000 or $139,000 for joint filers).  If you meet these qualifications then you can claim a lifetime learning credit.