It is my understanding that the Windows operating system (windows 7 to be exact) must come with a C compiler because the OS is written in C. How can I access such a compiler to compile my C source code?
Not necessarily - You'd just need the necessary libraries - like libc or mscrt if they are dynamically linked, or just suitable runtime support for the binaries you are running . Even many linux distributions do not come with compilers built in (I often end up installing build-essential for Ubuntu when I need to compile packages for example), so expecting windows to have one built in is not correct.
You can run compiled files without a compiler on the system. Its interpreted languages that often need the language tools installed to run - like Java or Python
If you must have a compiler, there's two fairly common options - the open source mingw or microsoft's visual studio - there's a free version called visual studio express. Install and configure one and carry one.