Have a wallet on your own webserver, this server as locked down as possible. Have a button on that server that can only send to one address
that already exists.
I'm not aware of anything that fulfills the OP's requirement of running on your own webserver and more or less being made for that specific purpose - a quick search didn't turn up anything either.
If you could please point me towards one project that facilitates that, I'd be very grateful.
I like the idea very much and think such a thing could be quite useful!