I'm not certain about the current requirements, but you used to have to provide proof of US citizenship, as well as proof of membership in a CMP-affiliated club or state organization, as well as any state requirements like the Illinois FOID, etc. When I first started buying from CMP (mumble-mumble) years ago, a copy of your voter's registration was accepted as proof of citizenship, but that was deemed insufficient at some point, meaning you had to send a copy of your birth certificate or passport.
In short, I doubt that your LE ID will be considered sufficient by itself.
You should be able to find the current requirements at thecmp.org/cmp_sales/rifle_sales/eligibility-requirements/.