If I understood you correctly, you need a system on which a user goes to a web site and then this system sends a message. However the user can decide which method to use to deliver the message: e-mail, Push notification, SMS, etc. AND also decide to which app to send it.
And you need this system, so your native apps accept it. In short you need and SDK for this.
If that's right, the close ones available for some
of these tasks are QuickBlox
. However none of these do all the things you request.
That said, the first caveat I see, is that PUSH notifications on phones are on a per app basis. It could be possible to make a broadcast PUSH messaging system, but in the end, the certificates for each app must exists.
For the SMS, one needs an SMS gateway, not impossible but certainly not free. Last but not least, In-App messaging and In-app inbox is sort of a WhatsApp clone. Completely possibly technically, but it does has it complexities, specially if you scale it up.
So, as you're seeing, each feature is a project in itself. We might be able to make all these for you :)
Best of lucks!