So, I'm quite inspired by the two answers above. My boot file is not a vuex file, there is no way for me to make an action. Instead, I made a function that can be called when a site is chosen and exported it. Here is the function I implemented to change the variable in my boot file.

let changeBasepath = (basepath) => {
    baseURL = basepath;
    api = axios.create({ baseURL: baseURL });
};

My boot file after the change looks like this:

import { boot } from "quasar/wrappers";
import axios from "axios";
import { LocalStorage } from "quasar";

import { Platform } from "quasar";

let baseURL = LocalStorage.getItem("basepath");
let api = axios.create({ baseURL: baseURL });

let changeBasepath = (basepath) => {
        baseURL = basepath;
        api = axios.create({ baseURL: baseURL });
};

export default boot(({ app }) => {
    // for use inside Vue files (Options API) through this.$axios and this.$api

    app.config.globalProperties.$axios = axios;
    // ^ ^ ^ this will allow you to use this.$axios (for Vue Options API form)
    //       so you won't necessarily have to import axios in each vue file

    app.config.globalProperties.$api = api;
    // ^ ^ ^ this will allow you to use this.$api (for Vue Options API form)
    //       so you can easily perform requests against your app's API
});

export { axios, api, changeBasepath }; 

Then, I imported the boot file and implemented the function. Here is the script chunk of my vue file looks like after the change:

<script>
import { LocalStorage } from "quasar";
import { changeBasepath } from "boot/api"; // <-- IMPORT IT

export default {
data() {
    return {
      companyList: [{
            id: "dev",
            site: "https://dev.mycompany.com",
        },
        {
            id: "demo",
            site: "https://demo.mycompany.com",
        },
        {
            id: "sandbox",
            site: "https://sandbox.mycompany.com",
        },
    ],
      loginData: {
        username: "",
        password: "",
        companyid: "",
      },
    };
  },
  methods: {
     checkCompanyId(payload) {
      let temp = this.companyList.find((o) => o.id == payload.toLowerCase());

      if (temp) {
        LocalStorage.set("basepath", temp.site); // <-- Here is the setting of the basepath
        changeBasepath(temp.site); // <-- IMPLEMENT IT
        return true;
      } else {
        return false;
      }
    },
     submitForm() {
        const CompanyId = this.checkCompanyId(this.loginData.companyid);

        if (CompanyId) {
           this.loginProcess(this.loginData);
         } else {
           console.log('Company ID can't be found!')
         }
      }
    },
  }   
}
</script>