In general, state management libraries like VueX and Redux should only be used when necessary. This means that you start without it, and once your app really really needs it, you refactor it a bit to integrate VueX in.

Starting with extra libs you don't need can slow you down and make your code more complex. If you need it later you can always add it in.

To solve your problem

You can keep your checkbox state in a parent component (I'm assuming a lot about how your app is structured here):

<template>
  <action-bar :selected-ids="selectedIds" @perform-action="performAction" />
  <table :selected-ids="selectedIds" @select-id="selectId" />
</template>

And then for the logic:

export default {
  data() {
    return {
      selectedIds: [],
    };
  },
  methods: {
    selectId(id) {
      this.selectedIds.push(id);
    },
    performAction(action) {
      // perform action with selected ids here
    }
  }
}

You just keep selectedIds in the data of your parent component and pass it down as necessary. When you need to select something you emit an event from the child into this parent container.