A 64-bit operating system isn't mandatory on 64-bit hardware, it just means you aren't using the machine to its full potential (and severely limiting the amount of RAM you are able to use). You also won't be able to run 64-bit applications.
With respect to the drivers, if HP's website isn't of any help, I usually identify the chip/controller in question and look up the model of that chip. Generally, manufacturers will have generic drivers you can use. For the network adapter, you should be able to locate the controller somewhere near the Ethernet port on the motherboard, it's likely going to be an Intel, Broadcom or Realtek.