Demo

HTML

<script>function 
myFunction() {
  var x = document.getElementById("myDIV");
  if (x.style.display === "block") {
    x.style.display = "none";
  } else {
    x.style.display = "block";
  }
}
</script>
 <h1 className='title-caption'>Get started today!</h1>
                <div className='email-newsletter-container'>
                    <p className='sign-up-caption'>
                        Sign up for our newletter for more information
                    </p>
                    <input
                        className='email-input'
                        placeholder='Enter email address'
                        onfocus="myFunction()"
                        
                    ></input>
                    <br />
                    <button style="display:none" className='submit' id="myDIV">
                        Submit
                    </button>
                </div>

CSS

.email-input {
        margin-top: 20px;
        height: 40px;
        width: 400px;
        text-align: center;
        border-radius: 5px;
        border: none;
        outline: none;
        position: relative;

        &:focus + .submit-button {
            display: inline-block;
        }
    }

    .email-input:focus + .submit-button {
        display: inline-block;
    }

    .submit-button {
      
        position: relative;
        border-radius: 20px;
        width: 90px;
        margin-top: 15px;
        padding: 10px 10px 10px 10px;
        font-family: 'Trebuchet MS', 'Lucida Sans Unicode', 'Lucida Grande',
            'Lucida Sans', Arial, sans-serif;
        background-color: transparent;
        border: 2px solid white;
        color: white;
        display: none!important;
    }