I don't think they can/do unless CIC suspects any fraud. I quote from
Canada Gazette archives here:
"There is currently no authority under the Citizenship Act or the Citizenship Regulations (the Regulations) to enable CIC to collect the SIN for the purposes of exchanging information with the CRA."
But there are amendments to the Immigration and Refugee Protection Regulations (IRPR) waiting to come into force that will authorize CIC to collect the SIN and to verify information with the CRA. Until then, CIC will continue asking for NOA's from applicants where they need to prove income.
I'm just stating the facts here. In case of NOA's required/recommended, I would advise to give it to CIC as any LVO (except VFS) would fall under its jurisdiction.