Some of the comments in the petition page (https://www.change.org/p/sadiq-khan-declare-london-independe...) are interesting:
"Even though the Houses of Parliament are situated in London, Londoners are only able to elect around 11% of MPs in the commons; Londoners have no say in how the vast majority of MPs are elected, yet they are still subject to their laws."
Yes, as they are subject to the laws of the entire country. The location of the Houses of Parliament has nothing at all to do with this. Plus, are we talking about the City of London or Greater London? The "11%" number essentially comes down to area boundaries and what is and isn't considered "London".
"London has no control over its borders while it remains a member of the UK."
What borders? London is a single city in a country. By definition the type of borders they are referring to don't exist, as it's not a country boundary.