Autocomplete.vue

<template>
  <div>
    <slot></slot>
  </div>
</template>

<script>
  export default {
    name: 'autocomplete',
    provide() {
      return {
        autocomplete: {
          setInput: (input) => {
            this.input = input
          }
        }
      }
    },
    data() {
      return {
        value: '',
        input: null
      }
    }
    methods: {
      onInput() {
        // Use this.input here to do stuff
      }
    },
    // Other properties... redacted for example purposes
  }
</script>

AutocompleteInput.vue

<template>
  <input v-bind="$props" v-on="$listeners" ref="input">
</template>

<script>
  export default {
    name: 'autocomplete-input',
    inject: ['autocomplete'],
    mounted() {
      this.autocomplete.setInput(this.$refs.input)
    },
    // Other properties... redacted for example purposes
  }
</script>