I never heard of a booking form either, though it sounds like an in-house term for a document specifying the parameters of an outsourced job. A "booking" is a scheduled event, as opposed to everyday routines. So a booking form probably means your contract, or letter of agreement, spelling out specifically what you understand the job to be and the details of your remuneration.
Big corporation or small, you're dealing with real people here, who are making it up as they go along as much as anyone else is .... !
I'm guessing about this, so really the best thing is to ask them exactly what they require in a "booking form."

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Mary H. Ruth/Virtual Writing & Communications
http://www.writingVA.com