Unfortunately, no. Screaming Frog is not capable of this. If you're comfortable using a terminal you could use OpenSSL though.
Here's a command that will verify the certs and pull the expiry date:
echo "Q" | openssl s_client -servername [domain] -connect [domain]:443 | openssl x509 -noout -dates
Example output:
~ % echo "Q" | openssl s_client -connect example.com:443 | openssl x509 -noout -dates
depth=2 C = US, O = DigiCert Inc, OU = www.digicert.com, CN = DigiCert Global Root CA
verify return:1
depth=1 C = US, O = DigiCert Inc, CN = DigiCert TLS RSA SHA256 2020 CA1
verify return:1
depth=0 C = US, ST = California, L = Los Angeles, O = Internet\C2\A0Corporation\C2\A0for\C2\A0Assigned\C2\A0Names\C2\A0and\C2\A0Numbers, CN = www.example.org
verify return:1
DONE
notBefore=Mar 14 00:00:00 2022 GMT
notAfter=Mar 14 23:59:59 2023 GMT
If I was in your shoes, I'd probably write a shell script to gather this data for each site and run it on a cron.