Civicrm webform to edit membership custom fields
Absolutely possible with Webform CiviCRM! A few thoughts:


upgrade to CiviCRM 5.3.1 or above (a security update)
by giving members logins or by using the checksum token you can make it such that their all ready exisiting in the database details (Name, Address, Membership Type) are displayed to them - you decide if you want them to be able to update those
of course in addition to filling out the blank Custom fields on the Membership;
look to enable the custom fields on your Membership in the CiviCRM tab for your webform -> Membership section (on the left) -> at the bottom of it. Here is an example: that Checkbox ALM is a CiviCRM custom field on the entity Membership in this example. Look here to expose your custom date fields to your Webform.