I'm sorry, but I don't think you mean specifically "Legitimists" (unless you're here referreing to Napoleon III). As a movement, legitimism appeared around 1830 with the Louis-Philippe Restauration.

If you're talking about the Royalists as a whole, then you must remember that there is a whole varitey of them, different groups with different aspirations. You've got the Ultra-Royalistes, favorable to the renewal of the Ancien Régime, the Constitutionalists, leaning towards the Revolution...

So we have to know more about your question to answer you fully. To give you a short answer : it's a movement, a process of "normalization" of the Royalists in the new political system.

At the beginning of his reign, there was a trend of hope among royalists that Napoleon could be talked into a restauration of the monarchy. Louis XVIII even started negotiations in this sense after the Coup d'Etat of the 18 Brumaire (9 novembre 1799), but they fell short.

One of the likely reason why these negotiations failed is the assassination attempt of the 24 december 1800. This what not the first attentat against Napoleon (see the Daggers Conspiracy), for example, but as the plotters have been revealed to be Chouans Royalists with links to the former Vendéens, it did have a significant importance.

It led to the reaction of both the stick and the carrot : harsh repression for the royalists involoved in the plots, but openness for those who want to join in the new political apparatus. It becomes visible around 1806, and obvious in 1810, which the year Napoléon I marries Marie-Louise (the grand-niece of Marie-Antoinette!).

But even from within the political institutions, the Royalists do not act. This is linked to a huge constant in the Royalists movements in post-Revolutionary France : they are always divided. What will become the Legitimists, the Orléanists, the constitutionalists already germ in their own respective movements. DUring the reign of Napoleon, the main dissent is between "moderate" and "ultra".

[please note that the last paragraph is more of a political science argument rather than a historical one, so don't take it for granted, some historians could very well -and rightly so- disagree with me here].

If you want to go further in this topic, I'd recommand this book :

Jean-Paul Bertaud, Les Royalistes et Napoléon. 1799-1816, Paris, Flammarion, 2009. Bertaud being one of the most famous French historian of the Revolution in activity.

Hope this helps !