How to set a $_POST var value as default value for a textfield webform component?
Webform 4 no longer uses so much of its own custom code.

It now uses drupal tokens like you would expect anywhere else.

Under the text area is a link "Browse available tokens." that will give you a javascript token browser to help you find the tokens you want.

You can access submission tokens like this: [submission:values:FIELD_KEY]

The special tokens you are after though like %post are no longer supported in webform 4.x.

For more information see https://www.drupal.org/node/1001798

Your best bet for such token support would be for them to be added to the main token module or another third party token module in the global token space.

$_GET tokens have an issue https://www.drupal.org/node/550164 however I don't think there is one for $_POST at this stage.

Alternatively you can write a hook_form_alter() in a custom module and set the default value there using $_POST.
That would be your easiest option I would say.