I have a similarly old hp laptop on which I run mint13 and debian7, it's a compaq 6220 which runs wireless fine.
Zorin is really meant for people who want GNU/Linux to look like Windoze but it should work on wireless.
You need to know what chip is used for your wireless hardware to get the drivers/firmware to run it.
Run lspci in a terminal and post the results, or without quotes run:
"lspci | grep wireless"
Hardware, particularly wireless stuff, can be a pain because "linux" is not paying the money when decisions are made.
Ideally you checkout the hardware first but if all else fails wireless wise you can use a USB that acts as an antenae and is recognized by the kernel.