Gifted a home in 2001. Cannot find paperwork for cost of home built in 1971. Will fair market value on real estate bill work?

So the problem is, if you are audited, the IRS will not give you any basis you can't prove.  You may need to report the sale as if the basis was zero, and take a tax hit.  The FMV as of 1971 (the build date) might be a fair estimate but you would have to hope that if you get audited, the auditor is in a good mood.  I'm sure you can at least prove the cost of the land from your county real estate records.

Also note, if the mother in law owned the home jointly with a spouse, and the spouse died before 2001, then the mother in law inherited a partial stepped up basis from her spouse.  That would mean that the mother-in-law's basis as of the date of the gift in 2001 would be 50% of original basis, plus 50% of the FMV on the date of her spouse's death.  That can be determined by a qualified appraiser using old sales records, so you may be able to prove at least part of the basis.

You can also increase the basis by any permanent improvements you made since 2001.  An improvement extends the life of the property or increases its value (different from a repair which only restores damage to as-was condition).  The improvement must still be part of the property; for example if you replaced carpeting in 2005 and again in 2015, only the 2015 improvements adds to the basis.

Finally, if you have lived in the home at least the last 2 years and never rented it, you can exclude up to $500,000 of capital gains from taxation if you are married filing jointly, so the fact that you can't prove a basis may not cost you anything after all.