Unfortunately most of these are not being manufactured any more either.
Renesas (that bought Intersil, that bought Harris) announced discontinuation of the 80C88 earlier this summer.
65C02 is still being manufactured by WDC.
The supply of older 8088 seems to be running dry as well.
There are a few open source BIOSes for x86 - GLaBIOS, my 8088 BIOS, the BIOS used on SBC-188, etc. All can be modified to run on various hardware.
The MS-DOS relies on just a few things, other than the x86 instruction set, such as some BIOS entry point addresses. John Coffman and I found these dependencies more than 10 years ago when working on SBC-188.
So hardware compatibility with IBM PC is not strictly required to run MS-DOS. The problems start once you'd try to run applications. Anything that bypasses MS-DOS or BIOS calls, which many MS-DOS applications do, especially to access display directly instead of using slow BIOS routines.
By the way, it would be an interesting experiment to try to find how does MS-DOS's IO.SYS work. While back, when MS-DOS was just released, the compatibility with IBM PC was not required, the idea was that hardware vendors would customize IO.SYS for their hardware, similarly to how hardware vendors would customize CP/M's BIOS.
--Sergey