The ones based on Java run on the JVM, which is always written in C :) The ones written in Lisp, well, at the bottom it's C again. I haven't seen a popular lisp yet with a meaningful amount of assembly in lieu of C.
I have some of these exotic Lisp Machines from the 80s which run no C code by default and where the C compiler is optional and written in Lisp. ;-) Obviously exotic and outdated. An alternate world.
> I haven't seen a popular lisp yet with a meaningful amount of assembly in lieu of C.
There are plenty to choose from, implement fully in Java like jikes, partially implemented in Java and C++ like OpenJDK, implemented in Java and C like MicroEJ, and lots of other ones to chose from.
Then there is project Metropolis on the OpenJDK, which targets to replace Hotspot with Graal, thus reducing the amount of C++ code on the runtime.
A bunch of languages can be used to write a Lisp runtime.