Yes, but only out of curiosity
Wizards do not need the internet, but they will occasionally use it out of curiosity or when researching about Muggles.
When you can summon any book, instrument or animal with a wave of the wand and the word Accio!; when you can communicate with friends and acquaintances by means of owl, fire, Patronus, Howler, enchanted objects such as coins, or Apparate to visit them in person; when your newspaper has moving pictures and everyday objects sometimes talk to you, then the internet does not seem a particularly exciting place. This is not to say that you will never find a witch or wizard surfing the net; merely that they will generally be doing so out of slightly condescending curiosity, or else doing research in the field of Muggle Studies.
Harry himself has never used the internet.
Has Harry ever used the Internet?
J.K. Rowling: No. He's not allowed near Dudley's computer and Dudley's the only one who's got a computer. He gets beaten up if he goes too near the keyboard. So no, he's never used the Internet. I use it a lot but not Harry. Wizards don't really need to use the Internet but that's something that you'll find out later on in the series. They have a means of finding out what goes on in the outside world that I think is more fun than the Internet. Could anything be more fun than the Internet? Yes!
Raincoast Books interview transcript, Raincoast Books (Canada), March 2001.