I believe one of the earliest places to come across a Quibbler dispenser is on the "Dumbledore's Army" level in Year 5. After you cross the river/stream, you will come to an area where you have to work on getting through a gate. On the back left area there is a shack and on the left side of that shack there is a Quibbler dispenser. Just walk up, press B....and then do that 24 more times and the achievement is yours!
Edit: A Quibbler dispenser looks like a newspaper dispenser. It's brown and has an actual Quibbler newspaper on the outside. You can see what it looks like in the actual picture of the achievement. I've tried to add a bigger picture of what it looks like below.