I got the wireless password, joined the network.
It didn't connect.
I stated this fact on social media.
The wi-fi icon in the top left corner was flashing intermittently, not staying solid as it does on other devices connected to the same network.
I thought I'd leave it for a little and have a bit of a Google around. But before I did, I checked my Twitterfeed.
Someone answered and said yes. Their partner had an iPad 2, had neer inserted a SIM and used on their wireless.
A thread on a Mac forum suggested I update the firmware on the router, but then why do other iPhones and iPads connect fine?
So I went back to my friend's iPad.
The wi-fi icon was solid. I tried to connect to the Internet and it told me I wasn't connected, even though the wi-fi icon was now solid.
I turned the iPad off and back on again. Once it was back to the Home screen, I joined the wi-fi network again. The icon started flashing again.
I opened the Settings of the wireless network I was connected to. In the DCHP tab, the IP and Subnet address were both flashing on and off. I'd never seen this before! The numbers would be there for a split second, then disappear again. Weird!
I noticed at the top the button Forget this network.
I didn't think I could do any harm, so I hit the button.
I searched for wireless networks, reconnected to the same network, put in the password and PRESTO! Wi-fi icon was solid straight away. IP Address and Subnet mask fields populated under DCHP.
I opened Safari and I was connected to the Internet.
So to answer this question: Yes. You can use an iPad on wi-fi without a SIM inserted.
I quickly jumped on Facebook and Twitter, apologised to #iPad 2 and updated my status to the correct answer.
Note: This was an iPad that was bought at JB-Hifi in Australia. It was not tied to any one network (like some iPads have to be used on AT&T or Telstra) and had not ever had a SIM inserted into it. Results may vary from iPad to iPad, depending on your country, your available national telecommunications services and your wireless network's hardware, firmware or sofware.