I get asked this from time to time. The simple answer is yes you can, just be aware of a couple of things.
If you opt for a class where the judge may ride, do not be offended if they decline. In the showing world, tradition calls for a treed saddle if showing.
Since the original was posted Thorn Saddle Pads have been accepted in the show ring for Shetlands and hard to fit ponies. The Solution and the Smart range is also seen on the show circuit, right up to Horse of Year show level.
For the Pony Club you may need to ask the DC if you are allowed to use the Thorn or equivalent pads, but I have seen more and more children/young adults using these.
You may be marked down for being in a treeless in the show ring in a less than traditional looking model. Some judges simply do not like them. If you want something traditional looking you will find that these days.
BD rules state that you can use a treeless saddle providing it is of a standard design, ie not western looking. If you want something traditional looking go for the Freeform, Barefoot or Sensation Dressage models.
We have had success in the Freeform Classic with the online BD accepted shows.