一、效果图

 二、思路：先请求一级select数据，通过选中的id请求二级数据，以此类推！

三、代码

<template>
  <div>
    <el-card>
        <el-form :inline="true">
            <el-form-item label="一级">
                <el-select v-model="c_id" @change="selectChange">
                    <el-option v-for="(item, index) in moviesData" :key="item.id" :label="item.name" :value="item.id"></el-option>
                </el-select>
            </el-form-item>
            <el-form-item label="二级">
                <el-select v-model="c2_id" @change="handlerChange">
                    <el-option v-for="(c2, index) in twoClassifyData" :key="c2.id" :label="c2.name" :value="c2.id"></el-option>
                </el-select>
            </el-form-item>
            <el-form-item label="三级">
                <el-select v-model="c3_id">
                    <el-option v-for="(c3, index) in threeClassifyData" :key="c3.id" :label="c3.name" :value="c3.id"></el-option>
                </el-select>
            </el-form-item>
        </el-form>
    </el-card>
  </div>
</template>

<script setup>
import axios from 'axios';
import { onMounted, ref } from 'vue';
// 存储一级数据
let moviesData = ref([])
// 收集一级的ID
let c_id = ref('')
// 存储二级数据
let twoClassifyData = ref([])
// 存储三级数据
let threeClassifyData = ref([])
// 收集二级的id
let c2_id = ref('')
// 收集三级的id
let c3_id = ref('')
onMounted(()=> {
    getFirst();
})
const getFirst = () => {
    axios({
        url:'https://www.fastmock.site/mock/11000304f7a663ddbba9d6f9ef67838b/api/api/first',
        method:'GET'
    }).then(res=> {
        if (res.status == 200) {
            moviesData.value = res.data.data
        }
    })
}
const selectChange = (idx) => {
    // 清空二级和三级的数据
    c2_id.value = ''
    threeClassifyData.value = []
    c3_id.value = ''
    axios({
        url:`https://www.fastmock.site/mock/11000304f7a663ddbba9d6f9ef67838b/api/api/${idx}`,
        method:'GET'
    }).then(res=> {
        if (res.status == 200) {
            twoClassifyData.value = res.data.data
        }
    })
}
const handlerChange = (e) => {
    // 清空三级的数据
    c3_id.value = ''
    axios({
        url:`https://www.fastmock.site/mock/11000304f7a663ddbba9d6f9ef67838b/api/api/${e}`,
        method:'GET'
    }).then(res=> {
        if (res.status == 200) {
            threeClassifyData.value = res.data.data
        }
    })
}
</script>

<style scoped>

</style>