TexasCHLforum.com

Well, found it, and it requires an FFL to do the check:

(2) a prospective firearm seller who does
not hold a current and valid federal firearms license issued
pursuant to 18 U.S.C. Section 923(a) shall arrange for a
person who does hold that license to conduct the federal
instant background check. A federal firearms licensee shall
not unreasonably refuse to perform a background check
pursuant to this paragraph; and

Here's the full bill text: https://legiscan.com/NM/text/SB8/id/1953393

Here is a little more clarification. Apparently the new law will require stores that do firearms background checks to do them for private party sales between two individuals. So, Walmart doesn't want folks walking in with guns and doing the transfer in their store.

https://thehill.com/homenews/state-watc ... on-july-22

Gonna have to dig into the law and see if NM made that a requirement of an FFL.

https://www.kfvs12.com/2019/07/03/walma ... ew-mexico/

(KOAT/CNN) – Walmart will soon stop selling guns in New Mexico after the state recently passed two new gun laws.