The concept of carte blanche goes back at least to Roman law, from which it was received into early medieval European civil and canon law. 1, 2 The basic idea is that someone with authority (A) can choose a delegate (B) to represent them, with "full powers" meaning that A agrees in advance to be bound by whatever B decides to do. B may well be given more specific instructions, and in any case the grant of authority does not mean that B is allowed to harm A's interests with impunity. B is only meant to do what A would have done, had A been present. The concept was known in the jurisprudence of France, as well as in the Roman Catholic Church, so Richelieu is certainly covered either way. In our days, we encounter the concept when ambassadors are described as "plenipotentiary", with the same sense of them being delegates of their head of state. But note that in the religious context, such delegation is still only ever intended to mean that B represents A's interests. If A is a bishop, B does not gain any episcopal powers; Papal legates are not Popes themselves; etc.
In his time, the term in French was usually applied to military orders, and indeed Richelieu himself granted carte blanche to his generals (or relayed that of the king) on several occasions.
We see nothing to do other than to give you carte blanche; the king leaves you free to attack anywhere that seems good to you. [...] For I repeat once more to you that the King leaves you carte blanche.
Nous ne voyons de loin autre chose à faire qu'à vous donner la carte blanche; le roy vous laisse libre d'attaquer telle place que bon vous semblera. [...] Car je vous redis encore une fois que le roy vous laisse la carte blanche. 3
Within the context of the novel, it is Richelieu's civil power which is being invoked. When he gives the letter to Milady (chapter 44), the idea is to give her immunity from the criminal law, although it is written in more general terms. Their conversation mentions the Bastille prison, but not purgatory. It's true that when our heroes come into possession of the letter (chapter 47), Aramis describes it as an "absolution" from all rules. This is an echo of his "confession" conversation with d'Artagnan in chapter 26, where the latter had joked "I'll give you absolution in advance" (i.e., before hearing the story), and Aramis responded "Don't joke about holy things, my friend". 4 Although Aramis instinctively uses religious language, there's no real sense that the letter involves Richelieu's religious authority, in the way that the musketeers make use of it.
If the cardinal had intended to write a letter granting absolution (or remittance of temporal punishment) to the bearer, he was exceeding his powers in several ways. It is true that as a bishop, he had the right to issue indulgences to people under his authority or in his diocese, just as the Pope could do so for people anywhere. We could argue that Milady and the musketeers all count as such, even if the letter itself is written in a more general way. But this document does not meet the well-established criteria for the validity of indulgences, or for penitence in general. Principally, you can't be absolved from a sin that you are intending to commit, but haven't done yet; you are not free from "attachment to sin". Compare Dante in Inferno 27.118-120 (Longfellow's translation of 1867):
For who repents not cannot be absolved,
Nor can one both repent and will at once,
Because of the contradiction which consents not.5
In the specific instance, Richelieu could never absolve Milady of murder before she actually did it, no matter what he wrote down in relation to her temporal punishment. Finally, sacramental absolution (for the other "bearers") cannot happen by means of a letter, as the priest and penitent are required to be in the same place for confession.
1. Consent, Coercion, and Limit: The Medieval Origins of Parliamentary Democracy, Arthur P. Monahan (McGill-Queen's University Press, 1987); section 3.1.C, p121ff on plena potestas.
2. Studies in Medieval legal thought: Public law and the state, 1100-1322, Gaines Post (Princeton University Press, 1964).
3. Lettres, instructions diplomatiques et papiers d'Etat du Cardinal de Richelieu, Avenel (Imprimerie nationale de Paris, 1853). Letter to Richelieu's nephew La Meilleraye, 17 May 1639: Volume 6, pp354-356. Found from a citation in Richelieu's Army: War, Government and Society in France, 1624-1642, David Parrott (Cambridge University Press, 2004), p136
4. My translation; original dialogue is "Et moi, je vois donne l'absolution d'avance, vous voyez que je suis bon homme. / Ne plaisantez pas avec les choses saintes, mon ami."
5. Ch'assolver non si può chi non si pente, / né pentere e volere insieme puossi / per la contradizion che nol consente.