While this article does talk about Avril Lavigne's reaction to the conspiracy theory, it doesn't actually claim that it's true - that she died in 2003 and was replaced by lookalike "Melissa." Therefore the conspiracy itself can't be proven true or false. The article serves to inform the public of the conspiracy and of Lavigne's thoughts in a recent Call Her Daddy podcast episode and an interview with Entertainment Weekly, the article's source.
To legitimize the existence of the theory itself, I searched "Avril Lavigne death conspiracy" on Google and found a variety of other sources covering the topic, including articles by The Guardian, BBC, and USA Today. The USA Today article featured the same quotes from the Call Her Daddy podcasts, which verifies that as a real source. It's clear that the theory exists and this article is covering a real story, but it's impossible to know if the theory is real since it is a conspiracy.