Do I need a special blender to make the smoothies on this site?

I have used a pretty cheap blender, an Oster Blender and a Ninja Blender.
While you need to be especially careful with a really cheap blender, it can be done.
One of the things I have liked about the Oster is how smooth the smoothie is.  One of the things I like about the Ninja is how much I can put in all at once, including ice.

For smaller blenders and cheaper blenders, put in just a few of the ingredients at a time, starting with the water.  Blend and then add more.  Once all of the ingredients have been added and blended, pour half into another container and put in the refrigerator.  Add ice, just a little at a time to the half still in the blender until it is cold, smooth, and the desired consistency.  When you have finished eating/drinking what has been blended and you are hungry again, after cleaning the blender, you can add the other half of the mixture and repeat what you had previously done with the ice.

For a large Ninja you can put everything in at once if you wish.  I generally put in all the ingredients, then blend, then add a little ice, then add the rest of the ice.