Matt,
I'm not personally familiar enough with Clausing lathes to know whether all models that could be called 13x36 have the same spindle nose taper nor what it would be. But I do know that typical spindle nose tapers for this general size lathe are 3MT or maybe 4MT. Anyway, if the spindle nose taper is generic and not proprietary like some brands, you need to ask for the part by spindle taper and collet type, and not by lathe brand and size. For example, if the spindle nose taper is 3MT and you want to use a 3C collet, then any 3MT by 3C collet closer adapter will work, whether it came from Clausing or not. Same with 3AT collets.