They’ll stick with a stable version that has the features they need until they have a good reason to move. That version will be one they’ve decided to ship themselves, or it’ll be provided by someone like Debian or Red Hat.
Most corporations are already using the likes of Nexus or JFrog Artifactory, regardless of the programming language.
No, people routinely write Rust with no third-party dependencies, and yet people do not routinely write C code that is memory-safe. Your threat model needs re-evaluating. Also keep in mind that the most common dependencies (rand, serde, regex, etc) are literally provided by the Rust project itself, and are no more susceptible to supply chain attacks than the compiler.