Our 2021 e-file was rejected, if I print and mail the return do I have to enclose a check or will the ACH payment I set up go through?

I don't think the ACH payment will go through since your return was rejected.  You'll have to enclose a check.

Many efile rejections can be fixed to allow you to resubmit the efile.   What was the specific rejection error code/message?

Even if you do end up filing by mail, another payment option is to pay by ACH at the IRS Direct Pay website.