Permissions required for using an app for a study
Yes, of course you should mention the name of the app. You have to provide details of your methodology. You should also cite the app (I see this commonly done by citing the app store page for it, but maybe your field has a different convention).
I have never heard of a researcher obtaining permission to run a study involving particular software. I have led or collaborated on about ten human-subjects studies involving a variety of free and commercial software.