I'm afraid it is not yet possible to transition issues through the customer portal by default. Currently, the only way to do that is by using a paid third-party add-on that extends the default JIRA Service Desk functionality (Service Pack for JIRA Service Desk).
We are tracking this request on JSD-40, which I'd advise you to add yourself as a watcher to, and to vote on it to increase its popularity.
I hope this helps!
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