Maybe? Ask the GM
This is an area where the game explicitly relies on the GM's judgment. There's no way to know which way a particular GM will rule until the player asks.
As cavaliers of the Order of the Dragon
dedicate themselves to a group of like-minded individuals, be it a mercenary company or a small band of adventurers. These cavaliers believe in loyalty and friendship, and are willing to lay down their lives to protect their allies.
…that description sounds to this GM as if such a cavalier should probably get the bonus himself. I can see another GM confronting me down at Hogwaller's for allowing that to work that way and that other GM saying that it "makes no sense" for an Order of the Dragon cavalier's challenge to benefit the cavalier because he's all about helping not himself but others. I'd counter with killing bad guys is helping others, even if the cavalier's alone. Punches would be thrown, and we'd spend the night in jail. After we're released, he'd challenge me to a PvP duel to settle things. I'd win, but, while we were dueling, his buddies would key my car.
Anyway, ultimately this GM figures the bonus is so little that it won't matter in the long run and would allow the cavalier to benefit from his own challenge, but a less generous GM might not. Steel yourself for disappointment. Or save up for some auto detailing.
(By the way, Paizo messageboard threads from 2014 and 2015 generally agree the Order of the Dragon cavalier should benefit from the challenge himself.)