If by "OS integration" you mean "write special code for each platform", then the answer will be negative.
As you mentioned, you need file system access and background processing. This combination is currently not available in a cross-platform manner.
If as a result of "OS integration" you meant "not waiting for Apple / google / ms to provide an opportunity," then the answer will be yes / possible.
All popular platforms have support for push notifications and support for background processing, if you encode it as each platform expects.
But access to the file system will be limited by the restrictions that the platform imposes on you. For example, in ios and win8 (phone), there is no need to write or read a file outside the native file structure of native applications. For security reasons, you cannot access the file system of other applications.
UPDATE:
The general template here is to release an application for each platform that you want to support. The application will register using the appropriate platform push notification service.
You will write generic server-side code to receive the data that you want to click on all your client devices. You will then call the appropriate push API for each platform you support, causing client devices to wake up and run the application that you provided for a response.
When the application opens, you will receive an application to contact your server and download the full data "push notification is just a call for your application"
This way, you can easily determine how your devices received data.
Each specific application on the platform must store data in its local storage and provide a way to share data using methods supported by the corresponding platform.
On iOS, it can be as simple as supporting the Open In .. paradigm. On W8Phone, you will have to publish data through one of the available “sharing agreements”, and so on for each platform that you want to support.
This is a common template at the moment. There are some reservations. On iOS, the application will automatically launch when a push notification is received. This means that your application will only download all data when the application is opened by the user.
The mobile application also cannot work in the background indefinitely. This means that after the application starts, you have a limited window for push notifications, which will be automatically processed by your application. After the allowed "background time". The application will close, and any push notifications will sound the device, but will not open the application until the user closes one of the notifications or opens the application directly.