I am just sharing a post from a very reliable source in this forum
@sylvain1 . This is pretty much what you would have to do and what to expect.
https://www.ourcommons.ca/parliamentarians/en/constituencies/FindMP
enter your postal code. After you click find MP, you will see the name and the photo of your MP. Click on the name, you will get phone number and email to join his office.
The best is to have the sponsor to make the first contact, especially if the sponsor has Canadian citizenship. Then, you cannot ask a Mp to have your application processed faster, at least not directly. You can ask his help to find if there is anything missing in your application or if there is any reason to explain why your application is not processed in the normal 12 months timeframe. You can also mention (if that is the case) that you tried several time to call the call center, but it's very difficult to get a line. You could possibly ask the MP if he/she can help in asking IRCC to at least issue the first stage of approval (SA and AIP), as IRCC should already have everything required at this point to make a decision on that.
There is no guarantee it will help, but it might be worth trying.