Yes, it's perfectly possible to write your C standard library - I did exactly that for my OS. Writing the header itself is relatively simple, you just look up the applicable standard (C or POSIX or such) and write it. It's slightly more tricky to implement the standard functions in the header (fputc, printf, ...) though. I have written an (unfinished) article on the matter, but it is meant as a follow up to the OS-specific toolchain article:
http://wiki.osdev.org/Creating_a_C_LibraryIt's a lot of work creating your own C library. If you are willing to do it, I recommend making your kernel support program loading and system calls. Then you should set up a
OS_Specific_Toolchain (note that I am currently rewriting that article!) and begin construction of your own libc. I am currently documenting this process, stay tuned.