At my previous position, I was in charge of something called the “Speakers Bureau” to review and approve speaking engagements for our leadership (and sometimes other staff members). Speaking requests would come to a special email-box for that purpose, and/or to me directly from employees who may have heard about a speaking opportunity. I then sent the requesting agency a form to complete with all pertinent details (location, date, time of speech, topic of speech, how many attendees, whether the press would be there, among other things). I had a standing bi-weekly meeting with our chief of staff to review all speaking requests, and she would either approve them, deny them, or ask for more information in order to approve/deny the request. This requires intense detail-tracking, but is definitely possible to do. I kept a spreadsheet to track the status of all the requests, and just saved a new file each week so I could clear out the past speaking engagements. I also kept hard-copies in a binder that I took to the bi-weekly meetings.
If approved, the speaking request also had to be approved by our headquarters Office of Public Affairs, who kept track of all the agency’s speaking requests for various purposes. Once approved, I would email the requesting organization back (although I was in constant contact with them to inform them of what stage in the process we were at), and continue coordinating additional logistics like travel arrangements, and sending them the final presentation. While all of this was going on, we also developed the talking points and/or powerpoint presentations for the speaker–and that was a whole other process unto itself!
Few other things to keep in mind for speaking requests. One is the topic that they are asking you to speak on. You should ensure that it aligns to your organizations goals and mission, and that you are not endorsing anything unknowingly by the nature of what the requesting organization does (for example, a federal agency cannot endorse any political candidates and/or certain other oganizations, so you need to know who supports/sponors the organization that is asking you to speak). You also need to make sure that if someone else from your organization has spoken at the same event in the past, that you ensure that your talking points match (that you are not in conflict with something the other person said–sounds weird but it happens!).
I hope this helps, and please feel free to message me directly if you have other questions!