Possession is nine-tenths of the law, and by that barometer, the Crown Loyalists have a very good claim to press, as they possess Thelbane, the Black House, seat of kings, and with it the literal Throne of Chaos and all of its trappings and accoutrements.
In more concrete terms, the Crown Loyalists are those factions and persons who are supporting the claim of Azalyn, daughter of Ishmael, to the throne. Her principal backers are House Hendrake, (which she will rule as High Lady regardless of how the various jockeyings for the throne turn out) House Helgram, House Alars, and House Nurgal. The Free Cities of Al-Rassan, Brandenburg, and Venetia have declared for her as well, and it seems likely that House Jesby and the Free Cities of Coruscant and Mega-Tokyo will as well in the near future.
This is a fairly significant chunk of Chaos' political weight, and represents a decent amount of varied military prowess as well; House Helgram brings with it Lord Loewen, Chaos' greatest living general, and the other Houses, though decimated, are capable of fielding a decent combined force for him to command, especially if Azalyn appoints him her Marshal of Chaos. If House Jesby and the Free Cities Coruscant and Mega-Tokyo come to the Loyalists as well, they represent significant percentages of what remains of Chaos' strength at sea and high-tech prowess, respectively.
Additionally, the Loyalists possess many of the great Primal artifacts of Chaos; Hendrake, Helgram, Nurgal, and Alars all bear their own Logrusfoils, House Helgram holds a second one, that of House Darios, as does House Hendrake, that of, ironically, House Suroth. Alars also holds a Primal Blade of Balance, and Helgram brings with it the Dragon Spikard Hide of the Dragon.
A number of prominent Amberites have given backing to the Loyalists as well. Prince Touga (who bears the Spikard Claw of the Dragon, another of Chaos' great artifacts) has more or less assumed the position of Azalyn's chief of staff, the power behind her throne, and Prince Orien and the Princess' Random, Shori, and Michelle have evinced either a preference for or given some degree of support to the Loyalist position as well. Prince Aburatsubo and Prince Kiyoharu are, in addition to being Touga's sons, members of House Alars, and thus have the double loyalties of their fathers politics and their mothers kin pulling them towards the Loyalists.
It remains to be seen whether or not King Tylor will promulgate an official and public Amberite foreign policy in this affair.
Azalyn is the eldest legitimate child of Dios' eldest legitimate child, who, not incidentally, was King before her. This would seem to give her claim to the throne rock-solid primacy. However, she is not a legal adult, has not assayed the Logrus, and is unlikely to even be able to any time in the near future. This makes her claim problematic, at best. The College of Heralds will need to make a legal ruling to clarify matters, and seem as likely to rule for Suroth as for Azalyn.
The Crown Loyalists have yet to articulate a very clear political stance beyond wanting Azalyn to be the Queen. Given the makeup of the coalition backing her, Azalyn seems likely to propose conservative, status-quo policies similar to those of her father, in order to ensure their continued political loyalty, but this is somewhat speculative.