The formal documentation of this is to be found in the club By-Laws and in the Recommended Club By-laws as Section 13 "Method of Electing Members".

Go to the RI Website and click progressively through downloads, "Club and District Support" and "Club Administration" 

There is also the brochure 254-EN "How to Propose a New Member" which contains the membership proposal form.

Basically the process is for a member to fill in a proposal form, and pass this to the club secretary for Board approval and determination of Classification. The prospective members name is then discreetly circulated to member and if no objection is received within seven days, the prospective member can be asked to join the club and be inducted.