I had no problem with running Tiny Core Linux 12.0 on the one fitted with the 500MHz VIA Eden processor, however that was not the case with the other two.
In the past I've encountered the SiS 550 CPU which was fitted to the Relisys RWT205CE which booted and ran Tiny Core Linux. This time I was not so lucky but I put that down to the fact that this example was only fitted with 128MB of RAM. (Verify any error messages on boot)
The example fitted with the Vortex
Booting Tiny Core Linux brought up the message:
This kernel requires the following features not present on the CPU: fpu Unable to boot - please use a kernel appropriate for your CPU.
As well as the limited RAM, lack of FPU (Floating Point Unit) in the CPU you can also have issues with the instruction set. Specifically the early Vortex CPUs are missing the CMOV instruction.
In my searching of the web I did come across this Config file for an old Linux kernel for a MicroClient Junior.