So I recently beat the game, and I completely forgot to go to Vault 101 again for the Trouble on the Homefront quest; is it possible to get back into Vault 101 and get the medicine bobblehead? If not, what point would I have to restart from in the game in order to get it?

That bobble head is available at the start of the game. Not sure about trouble on the home front.

If you have the PC version , just add the bobble head with the console command ... otherwise ^shrug^

Yeah, I usually get it while I'm escaping, but I didn't want to go through the whole lengthy vault sequence so I just started from my oldest save *right* before exiting the wasteland. I feel like there should be some way to open up the vault without the distress signal, but I just don't know how to do so..

Well the wiki says its available during "Trouble on the Homefront"

So when ever that quest was closed out would be the point ... its not the same for everybody , you would know better when that was.

*EDIT* For some reason I was still able to start the quest; I had never heard the radio signal before, but I just went right outside the vault door, selected the emergency frequency, and got the quest. I don't know why it worked for me as it said on the wiki that you only had a short period of time to acquire and complete the quest, maybe it's because I have broken steel?