Multitasking and multiprocessing can be accomplished on computers that do not have a memory management unit (MMU) to provide virtual memory. There are many operating systems that support multitasking and/or multiprocessing for processors that do not have an MMU. I don't know when Unix utilized virtual memory.
There are other hardware requirements besides virtual memory that Unix needs to implement its multiprocessing features. Key is the protected or supervisor modes of the CPU, i.e. kernel mode versus user mode.
Are there current Unix systems that don't use virtual memory?
I assume that all modern versions of Unix utilize a MMU.
uClinux is a version of Linux that does not require a MMU and does not use virtual memory. But don't expect the same level of security between processes as with real Linux. It is an OS for embedded devices to run trusted application programs.
Does the C runtime even support that?
The C programming language is not tied to Unix or Linux. Nor does it require virtual memory. C can be used to program 8-bit microcontrollers. A runtime library is specific to a version of an operating system and a compiler. There are versions of the C runtime library for uClinix for processors that do not have a MMU.