No desktops that I know of come with "built-in" wireless; even if a desktop is ordered with wireless capability, it is an actual card inserted into one of the PCI slots.  Wireless was probably not even a configurable option on desktops in 2003 and was probably only considered optional on laptops.

You would need a wireless card to connect to a network wirelessly - something like this:

www.newegg.com/.../Product.aspx

You probably don't need wireless on this old computer. Just place it near your wireless router and connect with an ethernet cable. All wireless routers come with wired ethernet ports. You might even need to buy a $10 network card for the ethernet connection. Most computers that old came with phone modems for dialup.

This desktop is the main computer on the account and there is no option to change that. I want to receive wireless signal from this desktop to a laptop. Desktop is connected to a Comcast modem and efforts to connect the router have failed due to what Comcast says is lack of a gateway. Would router set up successfully if I insert a USB wireless adapter in a USB slot of the desktop? And would I need another similar adapter in the laptop?  I have no problem connecting to the Internet at a library wirelessly with the laptop.

You need to add an inexpensive switch (like www.buy.com/.../218839604.html) between the cable modem and the PCs.

in order for the laptop to access the desktop wireless you need to have a wireless router in the system. There should be no issue in adding a a wireless router on a comcast connection I have setup several for others and myself.