How does your wife file her business taxes? Is she self employed on Schedule C or files a separate Business return? The contractor should be doing the same way as her. He enters the 1099NEC on his schedule C. What do you mean automatically?
If your wife paid him $600 or more she has to give him a 1099NEC. Has she ever given one to anyone before? Does she need help issuing and filing the 1099NEC? You need to use the Online Premium version or the Desktop Home&Business program or the Business program to prepare 1099s.
You wife's business has the responsibility to issue a Form 1099-NEC to the contractor if your wife's business paid the contractor more than $600. That's where the responsibility of your wife's business ends with respect to the contractor.
It's the responsibility of the contractor to file his own tax return, include the income on Schedules C and SE and pay any taxes due. Schedule SE is where your contractor's tax return will show the calculation of his self-employment tax that covers his Social Security and Medicare taxes which will be included in his balance due.