What kind of phone do you have, and from what service is the Amber Alert being issued? I do not get such alerts on my phone, so I am guessing that you opted-in to an alert service with your local authorities or similar service. Our township has an alert option, as does the local radio station and some news and information Apps. Once you figure out where the alert originates, you can modify the settings. Finding the source is the key.
Some alerts I’ve opted in for, I’ve set them to have no ringtone or vibrate, just a SMS popup. I have very few Notifications options active – I’ve disabled popups and sounds from nearly all Apps.
My iPhone 5 has a Do Not Disturb feature that I’ve programmed to be on from 10pm to 7am, so it does not play sounds or light up the screen, but the so the few alerts I do get are there when I wake up the phone. I sometimes manually activate it while I’m at work, too.