These may sound like stupid questions, but they sometimes get overlooked..
Do you have the proper fee prepared? Some skins cost silver or gold.
You have atleast one rainbow jar in your inventory?
Are you putting the skin on the proper species? Pets can only use skins made for their species, such as unicorns use unicorn skins and can not use something like a puma skin.
Used to be that busy pets couldnt have skins applied, but im pretty sure that was changed.
If none of these are the cause, then there may be a bug going on.