The French Wikipédia explains this concerning this use of "qui";

Quand le pronom relatif « qui » est employé sans antécédent, il représente exclusivement une personne et équivaut au pronom relatif « quiconque » :

• Qui part à la chasse perd sa place = Quiconque part à la chasse perd sa place.

This doesn’t mean that you can always replace "qui" by "quiconque" in that contexte. This is not true, even though the meaning of expressions consisting of only "quiconque" and of expressions involving "qui" without an antecedent can mean the same. The pronoun "quiconque" is used for the three cases of occurrence, subject and COD and COI and so is "qui".

1

• a) Quiconque le dit ne peut pas savoir de ce qu’il parle.
• b) Blessez quiconque par une faute d’inattention et vous êtes responsable.
• c) Dites-le à quiconque et ils vous diront que c’est faux.

2

• a) Qui le dit ne peut pas savoir de ce qu’il parle.
• b) Blessez qui que ce soit par une faute d’inattention et vous êtes responsable.
• c) Dites-le à qui que ce soit et il vous diront que c’est faux.

You’ll notice dans in the b) and c) cases something else is present in the sentence: "que ce soit". This support for the pronoun is always necessary. This choice is common, but "vous voudrez" is also used and there are variants as for instance "vous voulez", "vous passe par la tête", "que vous puissiez imaginer", …

However those completing phrases are not used with "quiconque" (ngram);

Dites-le à quiconque vous voudrez.

A possible reason for "qui" and not "que", as I conceive this problem, would be that the longer form "qui que ce soit que vous voudrez" has been consistently shortened in "qui vous voudrez". Is not to be excluded another case of "emanant" grammar that finally had to be recognised.

qui is here as a pronoun that would stand in place of the person you which to call. Thus, it really is replacing a person and not an object.

Check this sentence:

Amenez qui vous voulez.

We expect you to come back with someone. You can translate qui by anyone: You can bring anyone you want.
For an object, it would become:

Amenez ce que vous voulez.

This time, we expect you to come back with an object.
Be careful though, the pronoun here is ce and stands for the object that you which. You can translate it by anything : Bring anything you want. For instance, if we replace the pronoun:

Amenez le ballon que vous voulez.

Check also that last one:

Amenez ce qui vous plaît.

Here, ce is still the pronoun, and it replaces that thing that you like. Le ballon for instance.

Why is there que with voulez and qui with plaît?

Let’s check these sentences:

Je veux le ballon

Le ballon me plaît

We realise immediately that le ballon is the COD of Je veux, whereas it is the subject of me plaît.

Further reading