I know in C shell, if I wanted to check if the inputted number was in a certain format, like a zip code, I can use the following code:
Now if I wanted to do the same thing in bourne shell script, I assume it would be something like this:
but this doesn't work when I tested it out. Does bourne shell support regex in if statement conditions?

Basic Bourne doesn't have regular expressions in the [ command, no. You can use case in this case (sic) though.

The case statement uses basic glob patterns, not full regular expressions. The expr command supports regular expressions, and many modern Bourne-compatible shells have extensions which add regex support to the shell itself.