Change Sent By in Drupal 6 Webform Module?
Depending on the Webform version, you can accomplish this  in one of two ways:

Webform 2

When you edit the webform, look under Webform mail settings for E-mail from address. As long as you have an email field attached to the webform, you can select that field under the Component option.

Webform 3

Click on the E-mails task underneath the Webform tab when viewing a webform node. Within the E-mail header details fieldset, you'll see an option to change the E-mail from address.





The one caveat: Drupal sets the Sender:, Errors-To:, and Return-Path: headers to the site's email address. You can, however, change these headers using hook_mail_alter().

If you wanted to make sure the email field is set to user's email address (for a logged in user), you should've noticed the User email as default checkbox in the email field's settings, which will accomplish this and will work with the setting I mentioned above.