If we have not received 1098-T, will we be penalized for not reporting this year and will we be able to file it next year?

 No, if you did not have qualified expenses, there is nothing to report. Specific Instructions for Form 1098-T states: You do not have to file Form 1098-T or furnish a statement for:

• Courses for which no academic credit is offered, even if the student is otherwise enrolled in a degree program;

• Nonresident alien students, unless requested by the student;

• Students whose qualified tuition and related expenses are entirely waived or paid entirely with scholarships; and

• Students for whom you do not maintain a separate financial account and whose qualified tuition and related expenses are covered by a formal billing arrangement between an institution and the student’s employer or a governmental entity, such as the Department of Veterans Affairs or the Department of Defense.

If you paid in Jan 2021 for classes in 2020, then 1098-T would be 2021 and there would not be a form for 2020. 

The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or deduction or possibly your student has taxable scholarship income. 

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)

You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T.