Umm, since it's on a website, no. You don't need Wi-Fi, just simply an Internet connected device. Basically, you just need to be able to connect to the Internet. Wi-Fi is one way, but there are other ways, such as an Internet-connecting extension to the device or something. You can access the site without Wi-Fi. You just need to be able to access it via Internet. Since you're on this website, that means you're connected!
However, to send, transfer and in general use the site for what it's for, you need Wi-Fi. This sending/receiving happens via wireless communications (e.g. computer to DS), meaning if you don't have wireless Internet connections like Wi-Fi, the site is pretty much useless.
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So basically, if you can access this site, you can access the Dream World as well. But to use it to communicate with your DS, you must be connected by a wireless connection.
