"burger" is an abbreviation for hamburger. My guess is this: a hamburger is a dish once popular in Hamburg, Germany. It is ground up fresh meat made into a patty seasoned with salt and pepper, and few if any other seasoning, that is broiled or fried. The dish became popular in other cities, and so if you wanted the dish in another city, you asked for a hamburger. If you were in hamburg, you just asked for the house specialty.
That said, a veggieburger would be either (1) a dish popular in Veggieburg, North Dakota, made from boiled vegetarians or (2) a hamburger flavored with diced vegetables in addition to salt an pepper.