It may help if you clarify what you consider natural and synthetic wood glues. Do you mean you want to be able to collect the ingredients in the woods for some reason? Is there a particular type of chemical or other component in wood glue you are trying to avoid?
Also, what are you glueing to what? Right now, I just know that you want a desktop and you don’t want to make legs. Does that mean you are setting this desktop on the frame from your old desk? Are you gluing boards together on edge? Are you don’t joinery? I’m pretty sure that the type of gluing is going to matter, especially if you’re trying to find a glue that has undergone minimal processing.