Who is flying the vertibirds?

Fallout 76 starts on October 23 2102. This is much earlier in the timeline compared to other Fallout games, for example Fallout 4 starts October 23, 2287.

So at the time of Fallout 76 the BoS has not yet been decimated. Important to note is also that the Brotherhood has several chapters on the West and East coast so chances are big that they have not been completely wiped out.

The BoS is also not the only operator of the Vertibirds, the Enclave also uses them.

In Fallout 76 specifically there are automated Vertibirds flying around, known as Vertibots.

Sources: Fallout Timeline Vertibird on the Fallout wiki