https://joomla.stackexchange.com/questions/22309/custom-fields-dont-render-plugin-shortcode

You need to create a layout override for the field. Go to Extensions => Templates => Templates. Select [YOUR-TEMPLATE] - Details and Files. Click on Create override and select Layouts => com_fields => fields. That will copy the com_field render.php into override directory of a template (path is:/templates/YOUR-TEMPLATE/html/layouts/com_fields/field/render.php). Go to Editor and select the html => layouts => com_fields => field => render.php.

Find the following string

<span class="field-value"><?php echo $value; ?></span>

and replace it with

<span class="field-value"><?php echo JHtml::_('content.prepare',$value); ?></span>

You can off course use field id to only to the override on fields that need it:

<?php if ( ($field->id == 'X') or ($field->id == 'Y') ) : ?>
    <span class="field-value"><?php echo JHtml::_('content.prepare',$value); ?></span>
<?php else : ?>
    <span class="field-value"><?php echo $value; ?></span>
<?php endif; ?>