According to CrackBerry.com, the problem is that the Storm2 is not connecting correctly to configured networks. CrackBerry reports:
I have a review unit handy, so I played with the WI-Fi on it for a while. As many users have noted, my review unit was experiencing Wi-Fi issues, too. However, rather than call Verizon tech support, I simply resent the service books and then rebooted the device. That cleared up the issue.
The issue at hand is when Wi-Fi enabled, the device is not able to fully transmit voice and data at the same time and the Wi-Fi logo stays greyed out, indicating that it is not fully connected to whatever WiFi service you set it up. The fix is to call Verizon and have them place you on a more expensive data plan ($45+ BES Plan), power cycle your device and resend your service books.
I was able to connect the Storm2 to two different Wi-Fi networks with no problems. You'll know you're using Wi-Fi and not 3G when the network name appears at the top of the device status bar, and you see the Wi-Fi logo where you'd otherwise see the EVDO logo.
Anyone who is experiencing this issue should attempt to resend their service books and reboot before clogging Verizon WIreless's tech support lines.
According to CrackBerry, Verizon Wireless knows about the issue and is working on a separate fix. There's no ETA on when the fix will be made available.