At my 30th year high school reunion we had a photo booth. That is, an actual booth that people sat in and had four quick photos taken by an automated camera system connected to a laptop that displayed the four images after they were taken. There was even a white diffusion panel with a light behind it above the camera. There was a box of props nearby with various hats, glasses, boas, frames, signs, etc. Hilarity ensued.
One of the owners of the booth is the spouse of one of my classmates. They did the entire thing for a set fee paid by the organizers of the reunion. They had a basic printer on site that printed a card on (5x7?) photo paper with the group of four images from each time a person or group hit the button in the booth and a logo for the event. There were options the users could select before they pushed the big red button: color, B&W, or sepia. The photo booth people also saved the images and posted them either on facebook or to a website (it's been a few years since the reunion).
Just like actual old photo booths, the emphasis was not on high quality portraiture, it was on fun!
However, I get the impression this is not the kind of "photo booth" you are asking about. Nowadays "photo booths" at events are usually just portable portrait studios. (A camera, a couple or three lights, and a backdrop.) It sounds like the organizer of the event wants to provide a print to each attendee as a "thank you" for supporting the event and the charity associated with it.
I do not have access to a printer myself so I would have to have them printed afterwards.
Here's the thing: The cost of paying your third party printer to ship individually will dwarf the cost of producing the actual prints. On the other hand, the time needed to sort and mark each envelope yourself will be staggering. I've been there and done that (one time was enough). If the order list is handwritten, there will be several with illegible names and/or contact information. Chasing those down will be even more time consuming!
With any volume of prints more than as few as a couple of dozen, buying a decent inkjet printer to have available at the event will be cheaper in the long run for both you and your client than getting physical prints sorted and delivered after the fact.
I don't think the organiser wants the photos to be public, as many guests (all women) consume a large amount of alcohol during the evening and are not always at their best at the end of the night (but still want their picture taken).
Based on the nature of your specific event, the need to print onsite is even more germane than would normally be the case. If you have control over the lighting and the backdrop, then there's no real reason why you should not be able to shoot JPEGS (or [raw + jpeg] to be on the safe side) and print jpegs on the spot.
Most photographers I've talked to who do such work:
- Charge a flat fee to the event organizer for shooting the event and providing one print (or more) per attendee. This is the majority of the revenue they will generate from the event. They provide up to the specified number of prints for no additional charge at the event only. Most are prepared to go about 10% past the negotiated number of prints before they cut off or start charging an additional fee per print and they price the original number accordingly. If you've already made 220 prints for a 200 print event, there's not much most clients will say when you inform them you've already exceeded the agreed amount by 10%.
- Have plenty of business cards on hand at the event to give to people at the event. The cards include links to detailed info regarding how to obtain any applicable passwords and how additional photos may later be viewed online, downloaded, or ordered. I know one photographer who prints such info on the back of the photo paper ahead of time.
- Post proofs or full resolution images at the photog's website. Access to the proofs can be password protected to prevent public viewing. Proofs may be very low resolution or may have large obtrusive watermarks over the center of the images. Depending on the fee negotiated with the organizers, either web-sized versions or full size digital versions of the images may be provided for download at no charge. This is where a photo hosting site such as smugmug, zenfolio, photoshelter, etc. comes in very handy. Password protection is already in the hosting site's infrastructure. Digital downloads that can be no-fee or sold at low overhead with no printing or shipping expenses is another advantage.
- Have the event organizers promote the links, including password info if applicable, to the photos again once they are posted. The organization should use normal communications channels for the group who hosted the event. Things such as posting links on social media outlets for the organization, announcing it at the organizations place of business, newsletters, emails, etc. (depending on what type of organization it is). If going this route you need to have extremely short turnaround (1-2 days after the event) getting images posted online.
- People can go to the website to select and download images, order prints, etc. All of the prints can be mailed directly from the printer to the address provided by the person placing the order. This will be expensive. With most photo hosting sites all of the prints can be held until a cutoff date and shipped to a single address. Then someone has to sort the images and match them to envelopes for each recipient. This will be very time consuming.