I think he formed a bond with a spren, just the spren itself wasn’t seen „on screen”. After all, we can’t be sure when exactly Szeth swore the Ideal, it could be while talking to Nale but didn’t have to (for me it sounded more like Szeth just telling him what’s he going to do not swearing actually), it could be when he told Dalinar that he’s going to serve him or somewhere in between. Anyway after that we don’t have that much Szeth’s POV and he’s kind of occupied with the battle so I guess he just met the spren after the battle or something like that. That’s just the simplest explenation of the situation and the simplest is usually the most plausible one.
However, it’s kind of weird that his spren wasn’t shown, as it could be just adresem as being there without thinking the pacing that much and it would be believable why it wouldn’t have a talk with Szeth or something like that as there was a battle going on. So maybe there is more to that, something we’ll learn through WoB or in the next book (it’s supposed to be about Szeth after all).