At the end of Words of Radiance, when Szeth dies and gets revived, Nale, Herald of Justice, gives him "a replacement Shardblade" which is actually Nightblood, Vasher's sword in Warbreaker. If Szeth really is being induced into the Order of the Skybreakers, then it seems that he would bond with a spren, and be able to summon the spren as a Shardblade. Since we don't know very much about the Skybreakers, it's possible that they can't summon Blades in this way (though Nale can), or something like that. Or, maybe, Szeth is a member of the Order but not a Surgebinder because he doesn't have a Nahel bond (unless we see him using Stormlight in Edgedancer; I can't remember). This could be the reason Nale gave Nightblood to him, but being able to Surgebind is a large part of being one of the Knights Radiant.
It seems likely that nobody will know the answer to this question, because it probably hasn't been explained and won't be until the next book or two, but my question is: Why doesn't Szeth get a spren to bond with and the ability to summon a Shardblade?