If you are using the default Opencart 3 theme, you can follow the below steps to add a search box in the menu bar.

STEP 1

Add the below lines in /catalog/controller/common/menu.php

$data[‘search’] = $this->load->controller(‘common/search’);

Before

return $this->load->view('common/menu', $data);

STEP 2

Add the following in /catalog/view/theme/default/template/common/menu.twig

<div class="col-sm-3 col-md-3 pull-right">
        {{ search }}
      </div>

Here is the full code

{% if categories %}
<div class="container">
  <nav id="menu" class="navbar">
    <div class="navbar-header"><span id="category" class="visible-xs">{{ text_category }}</span>
      <button type="button" class="btn btn-navbar navbar-toggle" data-toggle="collapse" data-target=".navbar-ex1-collapse"><i class="fa fa-bars"></i></button>
    </div>
    <div class="collapse navbar-collapse navbar-ex1-collapse">
      <ul class="nav navbar-nav">
        {% for category in categories %}
        {% if category.children %}
        <li class="dropdown"><a href="{{ category.href }}" class="dropdown-toggle" data-toggle="dropdown">{{ category.name }}</a>
          <div class="dropdown-menu">
            <div class="dropdown-inner"> {% for children in category.children|batch(category.children|length / category.column|round(1, 'ceil')) %}
              <ul class="list-unstyled">
                {% for child in children %}
                <li><a href="{{ child.href }}">{{ child.name }}</a></li>
                {% endfor %}
              </ul>
              {% endfor %}</div>
            <a href="{{ category.href }}" class="see-all">{{ text_all }} {{ category.name }}</a> </div>
        </li>
        {% else %}
        <li><a href="{{ category.href }}">{{ category.name }}</a></li>
        {% endif %}
        {% endfor %}
      </ul>

      <div class="col-sm-3 col-md-3 pull-right">
        {{ search }}
      </div>

    </div>
  </nav>
</div>
{% endif %} 

STEP 3

Edit /catalog/view/theme/default/stylesheet/stylesheet.css and findout following

#search {
    margin-bottom: 10px;
}

Set margin-bottom:0px

STEP 4

Remove the following from /catalog/view/theme/default/template/common/header.twig

{{ search }}

So your menu will be look like below