this.$http change to Vue.prototype.$http , don't forget import Vue

import axios from 'axios'
axios.get('url').then((res)=>{
    
 }).catch((err)=>{})

the this of your public method does not refer to an instance of VUE at all. It is a ghost if you this can find $http and store. Do you call your external js package directly without importing the dependencies you need? Your this here points to Request.
import XXXX from 'XXXX'
bring in what you need to reuse.

make up the foundation of js when you have time

this.$http cannot point to the vue instance.
solution: you can pass the vue instance as a parameter to the request function in, request (this)

function API (prefix) {
  this.prefix = prefix
};

API.prototype.request = function (url, options) {
  url = this.prefix + url
  ... do the request
}

API.install = function (Vue, options) {
  var api = new API(options.prefix)
  Vue.prototype.$api = api
  Vue.api = api
}

const RequestAPI = Request.bind(this)
// RequestAPI
RequestAPI();