I am a school teacher. Our payroll system changed mid year 2019. At that time due to a program issue no Social Security or Medicare taxes were withheld. How should I pay in the amount due? If on my tax return what form do I use? All that I have found do not seem appropriate for my situation?
If you're an employee, then it's your employer's responsibility to withhold and pay any required Social Security and Medicare taxes. If your employer failed to withhold the appropriate taxes from your pay, then it would up to them to resolve the issue and take corrective action.
If your employer has determined that your wages are exempt from Social Security and/or Medicare taxes, then you wouldn't owe those taxes and you couldn't choose to voluntarily pay Social Security taxes in order to get Social Security credits. Teachers in some school districts are exempt from paying Social Security taxes on their wages, and instead contribute to retirement programs separate from Social Security. I don't know if anything like that is involved in your situation, though.
You as an individual can't report and pay the employee's share of Social Security taxes on your tax return. The only way that you'd need to report and pay Social Security taxes on your tax return is if you're self-employed. In some circumstances, employers consider people who work for them to be independent contractors, technically making them self-employed. I don't know if that's involved in your case, but if it is then your employer should have issued you an IRS form 1099 showing the amount of non-employee compensation paid to you. You would then use a tax return Schedule C to calculate your net-earnings from self-employment, and a Schedule SE to calculate the amount of self-employment, or Social Security/Medicare, tax owed.