Can I add Wi-Fi to my Apple Watch

You cannot add Cellular to an Apple Watch that was not purchased with Cellular. The GPS only models do not have any cellular hardware.

WiFi, Bluetooth and GPS is included with every model Apple Watch.

If you need Cellular, then yes, you need to buy a new Apple Watch that has a cellular radio. You can trade-in the current Apple Watch or gift it to a family member

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

However, as long as your iPhone is nearby (same moderate sized wood frame constructed home), then the GPS only model will do everything the cellular model will do.

The cellular model will just allow leaving the iPhone at home, while you go for walk, run, bike ride, etc...

I have had a GPS only model for 4 and a half years, and I have never suffered. I answer phone calls, make phone calls, iMessage, get weather, tell Siri to remind me when I get home, start timers, get all may iPhone alarms, alerts, notifications on my Apple Watch, etc...

Then again, my Wife wanted the cellular because she does not want to carry her iPhone in her pocket during walks. So she has cellular and I have GPS only. We are both happy.