I am trying to understand if it is ever possible for Sitecore to load connection strings values from Azure Key Vault, not from web.config (or ConnectionStrings.config) file?
I don't think it's possible to get those at runtime by default with Sitecore OOTB. But like Mark Cassidy mentioned in a comment above, you could leverage this https://docs.microsoft.com/da-dk/azure/app-service/web-sites-configure#connection-strings if you are hosting Sitecore on Azure Web App (you could set them in your ARM Templates, from within the Azure portal or by PowerShell/CLI commands).
Regarding Azure Key Vault, what you could do is to store your secrets in it and use them during the PowerShell/ARM Templates deployment. One example of using this at deployment time could be found here: https://alwaysupalwayson.blogspot.ca/2017/03/advanced-sitecore-arm-templates.html. It was a POC, experimental and one year ago, but in case it could help.