I sort of reject that the Fallout TV show has Easter eggs hidden in it because it, as a whole, is the equivalent of one of those fancy Hotel Chocolat ostrich-sized patisserie collection bastards that cost 40 quid. However. Eagle-eyed viewers of the Fallout show noted that episode 6 gives you a number for Valt-Tec that you can actually get in touch with - 213-25-VAULT (or, 213-258-2858). Charges apply, as well as international codes if you're outside the US, which makes it 001-213-258-2858.
If you text the number you get a reply from Vault-Tec saying "The next available appointment is 33 weeks from now, please stand by!" (handily captured by X user FanaticalGuy cos my response hasn't come through yet). And then the significantly less immersive "Reply Y to get recurring marketing and other texts from FallOut", which is quite funny. There is speculation that this is just a reference to Vault 33, the vault where main character Lucy was born and raised. On the other hand, 33 weeks from now is November, the month when both Fallout 4 and Fallout 76 came out.