Objects of Experience
As noted by a number of authors, Kant's use of Gegenständ and Objekt is not entirely consistent, but generally speaking, the word Objekt is the more general of the two terms, and Gegenständ usually refers to specific instances of objects. That being the case, Gegenständ is used almost without exception when referring to the intuition of objects, objects of experience and, consequently, the objects to which the categories apply:
"In Cognition, its Application to Objects of Experience [Gegenstände
der Erfahrung] is the only legitimate use of the Category."
(Critique of Pure Reason, B146)
"Now all intuition possible to us is sensuous; consequently, our
thought of an object [Gegenstände] by means of a pure conception of
the understanding, can become cognition for us only in so far as this
conception is applied to objects [Gegenstände] of the senses."
(Critique of Pure Reason, B146)
"That is to say, the categories serve only to render empirical
cognition possible. But this is what we call experience. Consequently,
in cognition, their application to objects of experience
[Gegenstände] is the only legitimate use of the categories."
(Critique of Pure Reason, B146)
However, the following reference to the intuition of objects seems to be an exception:
"Apperception and its synthetical unity are by no means one and the
same with the internal sense. The former, as the source of all our
synthetical conjunction, applies, under the name of the categories, to
the manifold of intuition in general, prior to all sensuous intuition
of objects. [Objekte überhaupt]" (Critique of Pure Reason, B153)
The more general word Objekte is often used for the objects of concepts. Given that concepts refer to objects in general — i.e. as a general reference to various objects or to various experiences of a single object, the usage of Objekt is well suited to the idea:
"It is by means of the transcendental unity of apperception that all
the manifold, given in an intuition is united into a conception of the
object. [Objekt]" (Critique of Pure Reason, B139)
The following is a good explanation of the differences between Gegenständ and Objekt:
"A Gegenstand can be understood as that which is given in the
intuitional manifold 'prior to the synthesis of understanding and
independently from it' (B145) and thus as an object of experience
simpliciter — that is, an object that is experienced (because it is
given in sensibility) but not necessarily cognized by the
understanding. [...] It can thus be aligned with the 'blind'
intuitions that are without concepts at A51/B75. An Objekt, on the
other hand, is that which is more robustly 'objective' since it
reflects Kant's innovative standards of objectivity. In other words,
it can be characterized as that which results from the understanding
synthesis of the intuitional manifold and thus as an object of
knowledge sensu stricto — that is, an object of experience cognized
by the understanding." (D. McWherter, The Problem of Critical
Ontology: Bhaskar Contra Kant, p. 18)
Things in themselves
When Kant is speaking of objects as they are independent of our senses, he often distinguishes them by speaking of them as things in themselves rather than as objects. He makes it clear that the application of the categories to things in themselves is inadmissible. He refers to such applications as transcendental as opposed to the legitimate empirical application to objects of experience:
"A transcendental use is made of a conception in a fundamental
proposition or principle, when it is referred to things in general and
considered as things in themselves; an empirical use, when it is
referred merely to phenomena, that is, to objects [Gegenstände] of a
possible experience. That the latter use of a conception is the only
admissible one [...]" (Critique of Pure Reason, A237/B296)
However, Kant sometimes refers to things in themselves as objects. In the following, he speaks of the impossibility of applying the categories to them, referring to them as Gegenstande:
"If, therefore, we wish to apply the categories to objects
[Gegenstände] which cannot be regarded as phenomena, we must have an
intuition different from the sensuous, and in this case the objects
[Gegenstand] would be a noumena in the positive sense of the word.
Now, as such an intuition, that is, an intellectual intuition, is no
part of our faculty of cognition, it is absolutely impossible for the
categories to possess any application beyond the limits of
experience." (Critique of Pure Reason, B307)
Here's another example, in which things in themselves are spoken of using Objekt:
"In this case there remains a mode of determining the object by mere
thought, which is really but a logical form without content, which,
however, seems to us to be a mode of the existence of the object in
itself [Objekt an sich] (noumenon), without regard to intuition
which is limited to our senses." (Critique of Pure Reason,
A289/B345)
Can Kantian categories only be applied to objects given by the senses?
Yes. I don't think there's any real debate about that. Throughout the Critique of Pure Reason, one of the principle themes which Kant maintains is that the proper application of concepts is limited to objects of experience. If that weren't the case, all his arguments against the paralogisms in the Transcendental Dialectic would lose their force. He refers to them as paralogisms precisely because such arguments employ the transcendental use (A341/B399) of concepts which, for Kant, are illegitimate:
"Our purpose is to speak of transcendental illusory appearance, which
influences principles—that are not even applied to experience, for in
this case we should possess a sure test of their correctness—but which
leads us, in disregard of all the warnings of criticism, completely
beyond the empirical employment of the categories and deludes us with
the chimera of an extension of the sphere of the pure understanding."
(Critique of Pure Reason, A294/B350)
