> what is the libssl distribution called in a given Linux distro’s package manager?

I think you're going to need to know that either way if you want to run a dynamically linked binary using a library provided by the OS. A package manager (for example Cargo) isn't going to help here because you haven't vendored the library.

To match the npm or pip model you'd go with nix or guix or cmake and you'd vendor everything and the user would be expected to build from scratch locally.

Alternatively you could avoid having to think about distro package managers by distributing with something like flatpak. That way you only need to figure out the name of the libssl package the one time.

Really issues shouldn't arise unless you try to use a library that doesn't have a sane build system. You go to vendor it and it's a headache to integrate. I guess there's probably more of those in the C world than elsewhere but you could maybe just try not using them?