When I execute a Format Document
command on a Vue Component.vue file VSCode replace all single quoted string with double quoted string.
In my specific case
As noted by @attdona the Vetur extension includes prettier.
While you can change the prettier settings, as per the accepted answer, you can also change the formatter for specific regions of a vue component.
Here, for example, I've set Vetur to use the vscode-typescript formatter as it uses single quotes by default: