This is possible. When creating request, change the Reporter field to whom you're requesting on behalf of.
Or raise it as yourself and add them as Request participants.
To the best of my knowledge, the only way to do this would be if you were a Service Desk Agent, or Project Admin with agent access. You could then still do this in the customer portal when creating the request, or via the Jira main web portal by the method Edwin recommended.
But otherwise, I don't see a way for other (non-agent) users to be able to do this. In that case, I would recommend just adding a text comment to the issue that is created to explain to the Agent who this request is really for and ask the Agent to change the reporter value to the username/email in question.
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