I work as a high school teacher for a religious school (501c3). I made over $40,000, but they did not withhold federal taxes. Is this a payroll error or am I exempt?

No,  you are not exempt from paying tax, and it is unfortunate that you might have been led to think so.  You may owe, depending on your other circumstances like whether you have dependents, filing status, etc.    Going forward, either request that they withhold tax or pay in your own quarterly estimated tax yourself.    When you print out your 2022 return it is going to print out with four 1040ES forms that you could use to mail in quarterly payments during 2023.

When you say "they did not withhold federal taxes", does that mean there is just nothing (or 0) in box 2  or was boxes 4 & 6 also 0 or empty?

It's a mistake, you need to fix it.  Fill out and give them a new W-4.

You are not eligible to be treated as a "religious worker" unless you perform sacraments and other religious duties similar to a minister or pastor.  But, if they withheld tax in box 4 and 6, then they didn't treat you as a religious worker anyway.  It just sounds like they entered the wrong code for your federal withholding when they put you in the system.

Alternately, your withholding is correct for what you told them.  If you told them you were single with kids, or married with kids and your spouse had no job, then zero withholding would probably be correct.  A single parent with 2 kids filing head of household would owe no federal income tax on the first $50,000 or so of income.  So it sounds like you had more than one job, or you claimed too many exemptions on the W-4, or you have a spouse who works and you didn't take that into account.

Or possibly, you made a typo with your income when entering your W-2 in Turbotax.  Or possibly the W-2 is wrong, did you in fact have federal tax withheld from your pay checks that is not showing up in the W-2?

When you prepare a new W-4, and you are married with a working spouse or you have more than one job, you have to complete the form and calculations one time taking all factors into consideration, then file new forms at all the jobs.  For example, the instructions might say to claim married with 1 exception at one job, and single with no exemptions at the other job.  Form W-4 has 4 pages of instructions and worksheets, not just the front page you turn in.