The culprits are the couple. They were really at the professor's house for the carpet, which is a valuable Persian original. The alleged search for the cat and Ayumi's kidnapping were actually a ruse to hide the fact that it was really the carpet they were after; they chanced upon it by watching the professor's video site, in which the carpet was visible in the background. The photo from the cat was simply taken from Eri Kisaki's video of her pet cat Goro.
The evidence pointing to this solution is the fact that the carpet in which the Detective Boys found Ayumi was not stained by the spilled coffee that had soaked her clothes when the thieves overpowered her. Also, the couple was the only interest group who had also inspected the carpet along with the vase, while the others had paid attention to the vase alone.
