You are using innerHTML for every field.

Input field shouldn’t be using .innerHTML use .value instead and for select you should loop through the options not use .innerHTML

window.onload = function() {
    var quote_price_input = sessionStorage.getItem("quote_price_input");
    var message_to_customer = sessionStorage.getItem("message_to_customer");
    var select_price_option = sessionStorage.getItem("select_price_option");

    if(quote_price_input !== null) {
    document.getElementById('quote_price_input').value = quote_price_input;
      } else
    document.getElementById('quote_price_input').value = "";

    if(select_price_option !== null) {
        var sel = document.getElementById('select_price_option');
        var opts = sel.options;
        for (var opt, j = 0; opt = opts[j]; j++) {
            console.log(opt.innerHTML);
            if (opt.innerHTML == select_price_option) {
              sel.selectedIndex = j;
              break;
            }
        }
      }

    if(message_to_customer !== null) {
    document.getElementById('message_to_customer').innerHTML = message_to_customer;
      } else
    document.getElementById('message_to_customer').innerHTML = "";
};

// Before refreshing the page, save the form data to sessionStorage
window.onbeforeunload = function() {
    var quote_price_input = document.getElementById('quote_price_input');
    var message_to_customer = document.getElementById('message_to_customer');
    var select_price_option = document.getElementById('select_price_option');

    if(quote_price_input !== null) {
        sessionStorage.setItem('quote_price_input', $('#quote_price_input').val());
    }

    if(select_price_option !== null) {
        sessionStorage.setItem('select_price_option', $('#select_price_option').val());
    }

    if(message_to_customer !== null) {
        sessionStorage.setItem('message_to_customer', $('#message_to_customer').val());
    }
}