Whomever vs Whoever (possessive)

[Who(m)ever’s application is denied] will wish to re-register next year.

The bracketed element is best analysed as a noun phrase in a fused relative construction functioning as subject of the sentence.

The fusion involves the pronoun "whoever", which is simultaneously determiner and head of the whole noun phrase (hence the term 'fused'). It has to satisfy the case requirements of both the relative clause and the matrix clause in which the whole NP is functioning, and since the pronoun is head of the subject NP, there is a preference here for "whoever".

The verb phrase "will wish to re-register next year" requires a personal (human) head noun, which rules out the non-personal noun "application". This leaves only the pronoun "whoever" as a possible head, which makes sense since it’s not the application that will wish to re-register, but some unspecified person(s).

The meaning is comparable to the non-fused “[Any person whose application is denied] will wish to re-register next year”, where the relative clause “whose application is denied” is modifying the head noun person”. Here are the two constructions for comparative purposes:

[Whoever’s application is denied] will wish to re-register next year.

[Any person whose application is denied] will wish to re-register next year.

As stated by tchrist in another question/answer,

The relative pronoun takes the case of the function it serves in the subordinate clause.

Your sentence doesn't actually have whoever in a subordinate clause. In fact, the subject of will wish to re-register is the application itself: "Whichever application is denied will wish to..."

The correct pronoun here is whose: "Anyone whose application is denied will wish to re-register."

If you really want to use a form of whoever, then that sentence makes clear it must be the subjective form, not the objective, because the whoever is the subject of will:

Anyone whoever whose application is denied will wish to re-register...

But the version starting with Anyone whose is more felicitous, I think.