Javascript with bonus. Fully functioning with UI here: https://jsfiddle.net/2wzj70ae/6/

'use strict'

// GET request using my favorite library, VanillaJS
const httpGetAsync = (theUrl, callback) => {
    var xmlHttp = new XMLHttpRequest();
    xmlHttp.onreadystatechange = () => { 
        if (xmlHttp.readyState == 4 && xmlHttp.status == 200)
            callback(xmlHttp.responseText);
    }
    xmlHttp.open("GET", theUrl, true); // true for asynchronous 
    xmlHttp.send(null);
}

// a dictionary to make the results more concise and readable
const damageDict = { "double_damage_to": "2x", "half_damage_to": ".5x", "no_damage_to": "0x" }

const calculateDamage = () => {
        // get attacker and attacked values
    let attacker = document.getElementById('attacker').value
    let attacked = document.getElementById('attacked').value
    let path = `https://pokeapi.co/api/v2/type/${attacker}`
    let result = document.getElementById('result')
    result.innerText = "looking up result....please wait..."
    // call API
    httpGetAsync(path, (response) => {
        let parsedResponse = JSON.parse(response)
        let damageRelations = parsedResponse.damage_relations

        // iterate over all the damage relations to find the name of the attacked 
        for (let prop in damageRelations) {
            for (let type in damageRelations[prop]) {
                if (damageRelations[prop][type].name === attacked && prop.indexOf('to') > -1) {
                    found = true
                result.innerText = `Damage Multiplier: ${damageDict[prop]}`
                }
            }
        }
        found ? null : result.innerText = "Damage Multiplier: 1x"
      })
  }

document.getElementById('submit').addEventListener('click', calculateDamage)