If all the dependencies are from crates.io, as long as the Cargo.lock remains in place, this should not be the problem. If Cargo.lock is removed, there can be a problem if some crate was massively yanked, like here, but it should be easily solvable by just changing the version - it's very unlikely that the author will yank the crate without providing actual version.
If some of your dependencies refer to git, of course, the repository can be deleted by its author at any time. In this case, if you really need this code to be accessible, the simplest thing to do is to make a your own managed fork.