Assumption: We are talking about an internal initiative
As there seems to be little space for errors, I'd make sure I have clarity on the final deliverable and the key success factor(s). Best — prepare something like a “dummy result” briefly or its outline to discuss with the sponsor.
I'd write an introductory e-mail for the sponsor to send out, endorsing me on the one hand, shedding some light on the “why,” the required result, and timeframe (advised embedded picture of timeline / no attachments), and inviting the stakeholders to an early morning, 30-minute meeting ;-) Best, in the sponsor's office ;-)
During that meeting... I thought it would work best to do a short presentation (alignment), leave a moment for Q&As (individual expectations), and most importantly — ask the participants to suggest their preferred meeting slots for appointments (commitment). The latter would be the risky part. You could take a big piece of paper (say, a typical “brown paper”), draw day columns and hour slots, and ask the participants to put their names on them. In case of “I don't have a calendar with me” arguments, you could always say — “Please suggest several options, and provide me with your phone number. I will be happy to get in touch after the meeting.” You could then put this information in the minutes / embed in your e-mail, and send out for final confirmation /w deadline, followed by individual appointments.
I hope this will give you some ideas to consider at least :-)
Oh, and if you haven't got a recurring appointment with the sponsor scheduled yet, you might want to consider doing so.