Creating a part of C compiler, most frequently a working optimization pass, is a regular assignment. The entirety of C compiler is too large to give it as a student assignment.
Depends which ISO C version, C89 is quite doable, and clever students might dust off a copy of "A Retargetable C Compiler:
Design and Implementation" or "Compiler Design in C" as starting point.
Back on my day, high school students would learn C on introduction to programming classes, while 10 year olds were able to write Z80 or 6502 Assembly and some of them would go on to create the games industry we have today.
I guess education system went downhill since then, which I guess it is kind of true, given the poliferation of header only libraries, as if C was a scripting language.