I think what you're proposing for the objection is subtly inaccurate.
Early Critical Kant
First off let's call the position "early critical Kant" (Groundwork , Critique of Pure Reason and Critique of pure Practical Reason) to describe the view where actions are moral when they result for a will acting from reason and no other motive (for more on this, a very good source is Marcia Baron's Kantian Ethics Almost without Apology). Worded negatively, no action is a moral action if it arises from any cause other than a pure maxim of action (that can be universalized, based on reason, etc). One thing to note here is that every action seems to be a de novo free choice for morality. A strong basis for this is the "Third Antinomy" of Critique of Pure Reason that contrasts the apparent determinism of the world with the freedom of the will.
Another point to remember in Kant's account is that "the right action" is only the right action when done for the right reason. In other words, the question isn't simply: Did I murder that guy or not? but did I murder that guy for reasons of the categorical imperative? If I didn't murder him but this was for any other reason, then it doesn't count as a moral action. (This doesn't make it immoral to not kill for other reasons; just that those actions don't rise to the level of morality).
Moral Luck
Second, let's think about the contours of moral luck: It's the idea that whether or not we act morally is influenced by our background and resources. For instance, I'm posting on SE rather than slashing people's throats in part because of the upbringing I had.
Interface between Early Critical Kant and Moral Luck
Thus, just on this level, Kant's definition of morality and moral luck are at odds with each other. Is being at odds an objection? One type of objection is internal -- on a view's own criteria it fails due to incoherence, etc. Another type of objection is that a view doesn't match with reality.
I'm not sure this is either of those, because this seems to be a disagreement about what morality is. Kant's claim is not that people's circumstances seem to matter to their actions -- Kant's claim is that actions are moral only when they arise from a pure will acting in accordance with reason, i.e. circumstances don't matter. Moral Luck, in general, operates from a more popular definition where moral refers to the actions themselves and not the internal thought process behind that -- and it is precisely the claim that circumstances do matter, because morality is about right/wrong actions built on motives and circumstances.
Moral Luck can be understood as an intuitional challenge to the Kantian claim. This is part of Hegel's claim against Kant -- that many people live moral lives because they never win the lottery or live immoral ones because they were raised in terrible ways.
Late Critical Kant
To make things a bit more complicated, Kant's own view may have resources to accommodate this. In what we can call "late critical Kant," in the Metaphysics of Morals (not the Groundwork) and Religion within the bounds of Reason Alone, Kant presents several things that complicate the early critical account. In MM, Kant gives virtue an important role in developing character -- it's not clear there (at least from memory) how much he manages the problem of virtue vs. every action being a pure chance to do virtue.
In Religion, Kant sets up an interesting idea -- an initial choice between good and evil that then informs the latter choices of the individual. Failing to make the good choice, makes it hard (or impossible?) to do moral actions; the good choice makes good easier.
This seems to admit at least one influence of moral luck.
