conditional validation in webform
Take a look at https://www.drupal.org/project/webform_conditional 

Just make both e-mail &amp; phone mandatory, but only show the relevant field, based on what was chosen in 'best form of contact'.

The module takes into account the 'state' of the form. It doesn't force mandatory, even if the field is set to be mandatory, but is hidden due to a certain select option ( in your case, the 'best form of contact' ).