Unfortunately Reporter (Customer) only has access to Service Desk customer portal and does not have access to JIRA interface for the same project (unless he is also an Agent). So changing Permissions Schema will not help in this case.
Customer can only add comments to his issue and you can configure Service Desk Automation to perform some actions based on the comment. Automation for example takes care of transitioning issue from Waiting for Customer to Waiting for Support when customer comments on the issue is in Waiting for Customer state. So customer has to limit his ticket updates to comments, unless you want to configure automation to perform a transition to the same state with script in post function copying the customer comment content into some issue field. But then you would need a way to differentiate regular customer comment from comment containing edits with is very cumbersome.
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