"woke" and DEI are amorphous signifiers to begin with. They're stand-ins for what reactionaries hate about cultural niceties and workplace management strategies, but have ultimately little to do with what the words imply. But in Fallout TV the DEI is literal.
1. the populations of vaults 32 & 33 were selected for genetic predispositions to positive attitudes and strong work ethics, which makes them compliant subjects of the secret order of middle managers. So the population is "woke" for the purposes of being indoctrinated into Human Relations ideology, as envisioned by Bud. Everyone is nice and gets along to avoid the danger of any social conflict.
2. these "breeding populations" are also racially and ethnically diverse to achieve maximum genetic diversity for the long isolation before reclamation day. That's your Diversity.
3. the vaults themselves are organized as a commune where everyone has equal claims to resources and luxuries, even though Bud's Buds reserve dictatorial privileges for themselves in secret. That's your Equity.
4. members of the breeding population are brought in to the inner circle of the Vault's government to give the impression that they're participating in a democracy, when really it's a managed democracy that guarantees the election of a Bud's Bud to the overseer position every time. Woody and Reg are both native 33ers allowed to participate in the council, but they have no real power because they're being undermined by Betty, Steph, and whoever else may be a Bud's Bud. That's your Inclusion.
Finally, this "wokeness" and "DEI" culture is the basis of the model that Bud and the Vault-Tec executives envision for the future. The civilization that Vault-Tec wants to build after Reclamation Day is basically the realization of a perfectly managed bourgeois firm, where the subjects are ideologically indoctrinated into passively accepting the dictates of their ruling class masters forever. For all intents and purposes, this is a conservative culture warrior's nightmare.