Hi @Ludovic Lasry
Welcome to the LastPass Community.
You do not need access to the current email address if you are able to log in and access your Vault settings, as outlined in this support article.
From your LastPass Vault: select Account Settings
On theGeneraltab, underLogin Credentials, enter your new email address in theAccount Emailfield.
SelectUpdate. Result: A "Verify Email" prompt is displayed in your vault, and you are sent a verification email to yournew email address(also specified in the prompt).
You will then need to verify the code sent to the new email address.