The BCI is not in charge of applications for residence permits.
For all requests, we invite you to visit the web page of the Population Service (SPOP) of the Canton of Vaud (click here)
You will also find online (click here) the "Formulaire d'orientation de l'usager" (User's orientation form). This tool will allow you, by answering questions, to determine if you are eligible for a residence permit according to your situation. It also indicates the steps to follow to file a formal application.