To access your env variables in a Vue Component file in Laravel, you will need to prefix your variable with MIX_.
In the .env file
To take your case as an example, if you intend to use APP_URL as a variable in your .env file, instead of APP_URL, you should use MIX_APP_URL like:
MIX_APP_URL=http://example.com
In your Vue Component file
You then access the variable by using the process.env object in your script like:
process.env.MIX_APP_URL
Say today you set a property named proxyUrl and assign the variable as its value, in your script in the .vue file, the code should look like:
export default {
  data () {
        return {
          proxyUrl: process.env.MIX_APP_URL,
        },
  }
}
After that you should be able to retrieve the property in your vue template as normal like:
<template>
    <div>
        //To print the value out
        <p>{{proxyUrl}}</p>
        // To access proxyUrl and bind it to an attribute
        <div :url='proxyUrl'>Example as an attribute</div>
    </div>
</template>
Here is the official doc released in Laravel 8.x, in case you want to have a look.