The radio signal for this quest will appear sometime after finishing the Waters of Life quest by walking anywhere near Vault 101, even hanging around the Megaton are will allow you to receive it.
It is possible to glitch yourself out of this trophy by installing the The Pitt if you're not careful, credit to Just2Dirty for this info on how to fix it:
It's worth mentioning but the Trouble on the Homefront quest can bug out with the DLC.
If standing next to Vault 101 doesn't display the Vault 101 Emergency Broadcast in the Radio Signals, then exit Wasteland Capitals by travelling back to The Pitt via the Train (furthest north west in the wasteland) then coming back to Vault 101, this will re-sync your Radio Signals, now you will find it at the bottom of the list.
If you have the vanilla release then simply don't install The Pitt but if you have the GOTY Edition then one possible workaround for GOTY Edition owners would be to buy the vanilla release, uninstall the GOTY Edition, start a new game with no DLC installed and get the trophy that way.
If you choose to accept this quest then go with the good ending as it's by far the easiest to do. Once you enter Vault 101 you need to find Amata and ask her what is going on, to which she will explain that after you and your Father left the Vault descended into Chaos and split into two factions. One faction is lead by the Overseer whilst the other is lead by Amata.
To get the full happy ending for this quest you need to persuade the Overseer to stand-down by either convincing him that the Vault is doomed and won't survive much longer or you can hack the Terminal in the Security Office and get evidence that the Security Forces are planning an unauthorised assault against Amata and her followers.
The trophy will pop pop once you have left the Vault and the door has sealed behind you.