Hello! I'm currently in the process of doing the same thing (organizing an invitational for the first time), so I figure I'd share some of my notions. Although I might point out that these are in the process of field-testing, after all, this is my first one.
First and foremost, I think it's of utmost importance to get it approved from your state director; they can alter your course entirely (if you haven't already seen it, the invitationals page is at
https://www.soinc.org/play/invitationals, which contains some requirements for hosting an invitational). Once you at least get in contact with them, you are probably good to officially start.
On the topics you specifically asked about:
Outreach to other schools: Just getting it out there is a big deal. Once it's out there and people hear about it, it will start to spread (from what I've seen, at least). Making a basic website (it won't need to be anything super fancy) that gives your plan and posting it on the forums is a good start. The state website might also be willing to include some of your invitational info.
Convincing people to volunteer: As SilverBreeze mentioned, food and volunteer hours can be good incentives. Motivating students about the fun of Science Olympiad might also help. Try to bring your team into it, as they already know the fun of Scioly and might be willing to embrace it from a volunteer perspective. And definitely don't turn down offers to help, staff members at your school may be busy, but they might be really interested and might be willing to help out too. I'm not certain about this one, but I wish you the best of luck with it!
Teaching people how to volunteer: There is a basic volunteer training tool on
https://www.soinc.org/participants/volunteers as well as a large amount of Science Olympiad volunteer stuff. On
https://www.soinc.org/officials/event-supervisors, you'll find a large variety of event supervisor (and test writer) resources, including the Event Logistics Manual, which contains really helpful information for each event, as well as general event design. You'll also want to make sure that ALL volunteers have read the event rules. Past this, it's really up to you how far you want to take this. Lots of states have volunteer guides, you might be able to pick one you like and suggest your volunteers follow it (your state might even have one). Some that I have found particularly helpful are the Washington State (
https://sites.google.com/site/wasoevent ... guide/home) and UNM Regional (
https://stemed.unm.edu/sites/docs/WHAT_ ... O_KNOW.pdf) guides, although these obviously contain information that relates to their specific tournaments.
Realistically, how many volunteers would I need: This really depends. You should have someone on each event (although some people
might be able to take on two, especially if you can't find enough). For events that require timing and testing, you should probably have 2-3 (including the event supervisor).
Test writing: This is a difficult one. The resources above in "Teaching people how to volunteer" contain information on test writing, but linking volunteers to the test exchange would be a really helpful resource (I've written a few tests, and having some tests for inspiration was a lifesaver). Make sure they know not to underestimate or overestimate teams. The how to write a practice test wiki page (
https://scioly.org/wiki/index.php/How_t ... ctice_Test) has some good info for test writing.
Hosting the actual event: I can't say much about this one yet. Probably just making sure that teams know what to expect. The earlier you can tell them everything (the schedule, rule clarifications, how you plan to run it--online or not, whether or not there will be food, etc.), the better it will be.
Speakers/other things that you would enjoy at a closing/opening ceremony: This is one I really can't speak for. Most NM awards ceremonies consist of them reading off awards and sponsors and sending you away. I assume SilverBreeze's recommendation is a good one.
How to attain sponsors: My personal plan for this is to have teams pay a reasonable entry fee and then send out a lot of emails to businesses around town asking them to sponsor us. You might have a harder time with this, given that there are a lot of struggling businesses.
Sorry for the wall of text...
I hope everything goes well for you, and I wish you the best of luck in organizing this tournament!
EDIT: Apologies for any repeats of info from above. When I started making this, I had only seen SiverBreeze's reply.