to answer the primary question, yes you could, but pepperoni is already cooked and processed and is hard (dried),it's easy to eat in small thin slices, but not in big chucks.
You could try grinding it down again and find a way to bind it in a patty (egg?breadcrumbs?) but the final texture might be weird,
pepperoni is also usually quite strong in taste and making a whole burger out of it would be overwhelming flavour (IMO).
As someone (SnakeDoc) wrote in the comment, you could finely chop pepperoni and mix it with your burger meat.
pepperoni is more or less ground pork (or beef or mix of both) with spices and smoked.
As a substitute,
You could make a patty of ground pork and beef, add some pepperoni spices, including smoked paprika (or some liquid smoke) and use that as a burger meat.