Are there any organisms that lack ATP synthase?
As Superbest mentioned, substrate level phosphorylation (SLP) is alternative pathway by which ADP is phosphorylated to ATP. In SLP, phosphorylation is coupled with a highly exergonic chemical reaction. Fermentative organisms (both fully and temporally fermentative) organisms depend on SLP for ATP. 

Note: fermentative $\neq$  anaerobic

Fermentation can either be a temporary situation, depending on the growth substrate (for e.g. yeast) or the only option as in the case of amitochondriate organisms like Entamoeba. You can refer this book if it is available to you. See this article too. 

You can also see the wikipedia page on fermentation; it mentions the SLP steps.