I need Customer Community Users to have access to their Contacts. Is this possible? Customer Community licence does not support Sharing Rules etc.
The owner of a Customer Community User will own their Contact Record. There are ways of allowing them to edit and update their Contact Record information through their User Profile. You can create workflow or use a custom object that mirrors those changes through triggers to their Contact record. You could also potentially do it using Process Builder (I've not tried that route myself). There's no need to directly share the record with them. Instead use other objects or workflow to make the changes via their Community User Profile.