It exists in Greek and Latin, and probably in most other Indo-European languages—at least in the ones I know: Dutch, French, German, Italian.
In order to create an indefinite relative pronoun in Latin, one uses an extended form of the interrogative pronoun, doubling it, so quidquid instead of quid "what"; or one adds -cumque to the relative pronoun, so quodcumque instead of quod "(that) which". The indicative mood is used.
Catullus describes how Egnatius is always grinning, whatever happens, Carmen 39, 5–7:
... si ad pii rogum fili [if at the pyre of a loyal son]
lugetur, orba cum flet unicum mater, [people mourn, when the bereft mother bemoans her only son,]
renidet ille. quidquid est, ubicumquest, [he grins. Whatever it is, wherever he is,]
quodcumque agit, renidet. ... [whatever is happening, he grins]
In Greek, there are many ways to form indefinite pronouns. One can add a form of the interrogative pronoun tis to the relative pronoun hos, but there are other ways. The indefinite clause is normally in the subjunctive mood. An example from Xenophon, Memorabilia 2.3, 6:
ἄλλοις μὲν ἀρέσκειν δύναται, ἐμοὶ δὲ ὅπου ἂν παρῇ πανταχοῦ καὶ ἔργῳ καὶ λόγῳ ζημία μᾶλλον ἢ ὠφέλειά ἐστιν.
[he can be pleasant to others, but to me, whenever he is there, he is in all respects, both in deed and in word, harmful rather than helpful.]
Marchant (1923) translates: "he is pleasant enough to other people, but whenever he is near me, he invariably says and does more to hurt than to help me".
It is much easier to find an antecedentless (and postcedentless) indefinite pronoun when it refers to something adverbial (a time or place) rather than a thing or person. Both Latin and Greek are (far) more likely to add an antecedent than English or Dutch. I was able to find a Latin example, but not a Greek example. Nevertheless, I think the origin of our antecedentless indefinite clause lies in indefinite clauses that do have antecedents, as in Latin and Greek.
An different but interesting development in French is that of the word quoique, "what which", similar to Latin quidquid. It can mean "whatever"; but it has also developed a meaning "however", and even "although", as a full conjunction. That is, there is no longer a "what" to fulfil the function of a subject or object in the subordinate clause. La Harpe to Voltaire (s.xviii):
Quoique éloigné du centre de notre littérature, vous en êtes toujours l'âme et l'honneur. [However far removed from the centre of our literature, you are still/always its soul and honour.] Modifying the adjective éloigné.
Les écrivains sacrés, quoique remplis d'une lumière supérieure et infallible, s'expriment d'ordinaire d'une façon humaine et populaire; ... [The sacred writers, though full of a superior and infallible light, normally expressed themselves in a human and popular manner; ...] Modifying the entire clause (leaving out the finite verb as in English).
Indefinite relative clauses seem to be intimately connected with a concessive ("although") connotation in many languages.