is the vuetify time picker capable of setting a specified minimum time for 15 minutes?

You can try this

<v-time-picker
   v-model="time"
   :allowed-minutes="allowedStep"
   format="24hr"
   scrollable
   min="9:30"
   max="22:15"
></v-time-picker>

And you can add the allowedStep method as follows:

methods: {
  allowedStep: m => m % 15 === 0,
},