Phone must be on to sound the alarm .I don't know many silent alarm clocks. Doesn't matter if airplane mode is on or not for the alarm to function .
The only way I think this is possible is if you have the option to individually change the volume of the alarm ringer and enable the vibrate function. I use a Nexus 5X and I do have that option but it may not be available for all phones.
Manage your T-Mobile account