How do I specify the path of bash when trying to build
the kernel (2.4.20)? My bash is installed on /usr/local/bin/bash
but I can't do anything about this as I don't have admin rights.
I tried various workarounds like setting BASH as an env variable
containing the path and editing the CONFIG_SHELL variable
in the Makefile but none of them work. Keeps giving me the
same error message:

make: /bin/bash: Command not found
make: /bin/bash: Command not found
make: /bin/bash: Command not found
make: /bin/bash: Command not found
rm -f include/asm
make: /bin/bash: Command not found
make: *** [symlinks] Error 127

why are you compiling a kernel without root rights?

Edit the makefile to use your bash instead of the default one

Because I'm not root on the machine I'm using.

I tried setting CONFIG_SHELL to my bash path but it just
refuses to work. I also tried grepping and replacing around the
entire source tree for /bin/grep with /usr/local/bin/bash
but it still doesn't work.

you should not compile anything as root. Every step should be run as a regular user except for modules_install and install (when you actually need it).

In general, for standard Linux security philosophy think about it this way.
Never ever use root. If you are unable to do something try a different way. If you still can't do it then use root to do that step and that step only.

Willy nilly using root privleges when you don't need it can get you into serious trouble unless you understand it.

Feel free to completely ignore this warning.