The core issue was caused by the expiry of two intermediate certs in our cert chain. The root and end were valid, but the intermediate certs were invalid. These intermediate certs expire every 20 years (see: https://news.ycombinator.com/item?id=23360624 and https://twitter.com/dnsimple/status/1266729739915591681).
Since an intermediate cert failed, SSL still appeared to be functioning properly in web browsers (Chrome loaded https://codecov.io just fine), but other applications -- such as cURL, which our bash uploader depends on -- did not (see: https://github.com/codecov/codecov-bash/issues/309).
Many of codecov's uploaders rely on cURL in some form or another, thus the expiry of our intermediate certs led to many of uploaders malfunctioning, and users who attempted to upload coverage reports at this time were very likely not able to do so. We expect that this issue lasted from 12:01a CDT on June 1, 2020 to approximately 2:30p CDT on June 1 2020.
We resolved this issue by removing the expired intermediate certs from our chain and updating our certificate usage in our production environment accordingly. Therefore, no users should be experiencing SSL related errors or issues when uploading to codecov. If you are experiencing this or any other error, please let us know.