The door is a vinyl screen door. All vertical and horizontal members are vinyl.
For some reason, they used wooden dowels to assemble the joints. The dowels rotted and the door separated. I have replaced the dowels, but haven't found the right adhesive to cement the vinyl faces together (dowels alone are not tight enough to prevent sagging).

"I have a vinyl patio door whose construction included assembly
with wooden pegs."

Are the "wooden pegs" what are commonly known as dowels? Do they
attach wooden components to wooden components?

If so, why didn't you use wood glue?

If not, please describe the connection points in a little more detail.
Better yet, post pictures of the situation.

I've used super glue to build up trim profiles from vinyl trim, but I
was gluing vinyl to vinyl. I know that window/door trim is not
structural, but when I tested the bond, the vinyl broke, not the bond.

Can't say if that will help you since I don't exactly know what you
are trying to fix.[/quote]