Solution
Yes, the NI Application Web Server on cRIO-9045 supports HTTPS. Starting with LabVIEW 2024 Q3, TLS 1.3 is supported. For earlier supported versions of LabVIEW, TLS 1.2 is available. (TLS Support for LabVIEW - NI)
Regarding certificate expiration, the X.509 standard does not support indefinite validity. LabVIEW-generated self-signed certificates have a default validity of 10 years. While it is technically possible to use OpenSSL to create certificates with longer validity periods, this is not recommended. Additionally, most modern browsers will block connections to sites using self-signed certificates, as they are not trusted by default. (Using SSL certificates with the NI Application Webserver - NI)
If the cRIO communicates with an external Windows PC running the NI Web Server over HTTPS, the certificates used by the AppWeb server are stored locally on the cRIO.