[SalesForce] Can a community user edit their own contact record

For all cases where portal licenses don't permit you to do what you want, create a wrapper class of the fields you need, query the contact by id, and let the customer edit. Then take the wrapper and write back to the contact.

I've had to do this for the very same reason and it's very straightforward.

Also, I think I read somewhere that the customer community replacement for the portal fixed this issue, so there may be a non-code solution.

No this is not possible. When you create a customer community user, the user's role is determined by his/her contact's account owner's role.

This determines the level of access the customer community user has on the contacts of his account and other related records. If you change the account on the contact, then the whole roles, access and sharing or related records need to be revoked from the old account and need to be created in new account which is cumbersome.

I could find this documentation mentioning about the roles.

The available profiles for external users are limited to the Partner Community User profile or profiles cloned from it. Unlike a standard user, the partner user role is automatically assigned based on the account name. The partner user role is a subordinate of the account owner’s role. Therefore, all data for the partner user role rolls up to the partner account owner’s role. If you disable a partner user, the partner user role becomes obsolete, and the data no longer rolls up to the partner account role.