Should we have a federal 1099 form or Schedule K-1 (Form 1065) if the income was from making a sculpture for a private order? The customer paid 10000.

Your post is confusing ... what did the IRS actually question ?   Are they asking you to prove the income was recieved ?   What LTR or CP number (upper right corner of the notice)  did they send ?   

You won't receive a 1099 unless the purchaser was another business, and you won't have a K-1 unless the sculptor has organized their business as a partnership or multi-member LLC.

You need to work out what the IRS really wants. 

If your father-in-law is self-employed, he should have a schedule C to report his business income and expenses.  If he reported cash income of $10,000 on schedule C, the IRS may be asking for proof this was real income (there are a couple of tax frauds that, somewhat surprisingly, involve reporting more income than you legitimately received instead of less.)  If this was a private customer who does not issue a 1099-NEC, then you can send a letter of explanation along with a copy of the bill of sale, copy of the canceled check, business financial statements, and so on, to prove this was real business income.

If your father in law reported this as "other" income not on schedule C, that would be reporting it like a hobby.  The IRS may be wanting to verify what kind of income this really was--some people do sculpture as a hobby instead of a business, but probably not for $10,000.  He could reply with a letter of explanation and copies of the bill of sale, and explaining why he considered this hobby income instead of business income. Here are the factors involved,

https://www.irs.gov/newsroom/heres-how-to-tell-the-difference-between-a-hobby-and-a-business-for-tax-purposes