Are GNU coreutils SHA digest functions FIPSvalidated (in NIST's Cryptographic Module Validation Program)?

My Challenge
My project has a requirement that we use only FIPSvalidated modules to do anything cryptographic, including generating checksums for binaries. We've been using the SHA2 utilities provided by GNU coreutils for a long time to generate checksums. Now we need to be compliant. SHA2 comes from FIPS1802, and I can't find a validation program like NIST CMVP for FIPS180. I did see on some CMVP certifications that a Secure Hash Standard (FIPS180) certificate was also listed, so I had hoped to find GNU holding a certification for their implementation in NIST's CMVP database, but I couldn't find one.
My Question
Can anyone point me to documentation that shows GNU coretuils is validated for FIPS1802 compliance? If not, I'd love suggestions for where to look for an alternative provider. I did find that Red Hat holds a certification for an OpenSSL implementation, which also lists SHA3 and SHS certificates, Cert #3781, but that's for RHEL8 and we're on RHEL7.
Barring alternatives, is there a way to prove GNU coreutils's SHA2 implementations are "sufficiently secure"? I feel that question is asking for trouble and a whole lot of math proofs... The GNU website doesn't say much about SHA2. I read the source code for the utility here, but found nothing interesting.

I can't find a validation program like NIST CMVP for FIPS180.
You did, actually. CMVP covers compliance for all cryptographic mechanisms approved by NIST, including the hashes defined by FIPS 180. FIPS 180 is the definition of some hash algorithms. FIPS 140 is the standard that defines what it means to be compliant. All “FIPS compliance” is compliance to FIPS 140.
I did see on some CMVP certifications that a Secure Hash Standard (FIPS180) certificate was also listed, so I had hoped to find GNU holding a certification for their implementation in NIST's CMVP database, but I couldn't find one.
You looked in the right place. You couldn't find one because there isn't one.
FIPS 140 validation is expensive. You don't just do it on a whim: you do it because you have a clear business requirement. Furthermore, you can only validate one specific version of a program, so it's bad for maintenance (no bug fixing, no new features, no porting to a different environment, …) and it's bad for security (no bug fixing). So it's no surprise that very few programs are FIPS certified.
If not, I'd love suggestions for where to look for an alternative provider. I did find that Red Hat holds a certification for an OpenSSL implementation, which also lists SHA3 and SHS certificates, Cert #3781, but that's for RHEL8 and we're on RHEL7.
Several vendors have certified a version of OpenSSL, including RHEL 7, which does cover the NIST hash algorithms. So you could use that.
Barring alternatives, is there a way to prove GNU coreutils's SHA2 implementations are "sufficiently secure"? I feel that question is asking for trouble and a whole lot of math proofs...
FIPS 140 certification has nothing to do with security. In fact, it's detrimental to security, since it forbids applying security updates. If you want FIPS certification, use the certified version of OpenSSL. If you want security, use an uptodate system.
If you care about security, the software that calculates hashes is the least of your worry: of all cryptographic primitives, they're the hardest to mess up. What you use the hashes for is more important and can be susceptible to subtle pitfalls.