Just a set of words, or is there also a reconstructed grammar letting us speak in Common Proto-European?
Absolutely! In fact, one of the best ways to show that a language is Indo-European is through its grammar—words are easy to borrow, swaths of morphology, not so much. The Hittite words wādar "water" and ēd- "eat" were a good clue that the language was IE, but it was the inflections that clinched it.
is it only a set of words, or could we spoke this reconstructed language? (even if it's only a reconstruction)
Modern reconstructions are definitely complete enough to speak, though nobody's particularly fluent in it (because what would be the point?). But if you look up "Schleicher's Fable", or "The King and the God", you can find some texts scholars have composed in reconstructed PIE.
Schleicher's Fable in particular is a fascinating way to compare different theories, because it's become a sort of tradition now for anyone publishing a major new theory of PIE to show what the Fable would look like in their reconstruction.
If there's no reconstructed grammar, or not enough, does a conlang exist trying to mimic it? Meaning that it keeps the vocabulary, and invent a grammar.
Depending on your definition, all reconstructions of PIE are a posteriori conlangs! However, I don't know of anyone who's tried to flesh out PIE grammar for artistic purposes rather than as an attempt to explain the data.