App services run in isolation so no, not directly. You can of course copy to an intermediary if you have configured permissions as part of your app service and then copy from there. Alternatively if you have strictly an Azure solution then perhaps an Azure Function or something that has access to both app services and can copy from one to another.
But you said at deployment time so suddenly this doesn't make sense anymore. If you're deploying an app service then the files come from your deployment script. If that script wants to copy updated files then it would need access to the other file location (app service in your case) to copy from. This would be no different than copying from any other file location within the script but the script would need access to the app service, which is where the security challenges come in.
Personally I would just have the source app write to shared storage somewhere (perhaps a file server or something) and have your deployment script pull from that. This seems the most secure and allows for you to update the files even if the source app service isn't running.