Edit the Jira Service Desk request type

This is not currently possible in Jira Cloud. Jira Service Desk does not provide any APIs to change an issue's request type: JSDECO-36.

For server and are using Service Desk 3.3 or later, you can simple select the "Customer Request Type" field from the field picker

Change request type

If you are on a version of Service Desk before 3.3 you can still set it through Additional Fields following the instructions at - How to set request type using REST