Do rust and cargo allow for multiple interpretations of the same C header file across different objects in the same program? That's how C libraries are often implemented in practice due to preprocessor tricks, though I wish it wasn't normal to do this sort of thing.
However, some people use cargo's build scripts to build c programs, which then you can link into your Rust program. Support would then depend on whatever the person wrote with the script, which in my experience usually delegates to whatever build system that project uses. So it should work fine.
If the header files are consumed by C code that is then consumed by Rust then you'll have full support for what C supports because it will be compiled by a C compiler.