I've done similar shoots.
To print onsite you must have an assistant who is very familiar and comfortable with the software and the entire workflow. If you are busy enough to make a profit, you will not have time to consult with the client or assistant on workflow or image optimization during the event. Ideally the assistant also needs good people skills, to work with the customer.
Really just about any software will do. I've even seen folks using the free software that came with their $100 Epson inkjet. Just practice a little with it beforehand.
Limit the paper sizes you use to two or three (say 5x7 and 8x10/8.5x11) and only offer glossy paper. It's very expensive to try to stock a lot of choices.
Whatever the smallest size is that you offer, that's what you will mostly sell. If you offer 4x6, that's what 90% of your customers will buy. I stopped offering 4x6s. Too cheap to make it worth my while ($5 at the time). By offering 5x7 as the smallest, I probably lost a few sales, but my average sale per customer immediately doubled.
And, can you say "copyright infringement"? One time a lady who bought ONE 4x6 from me later bragged to me that she'd scanned it and used it for all her Christmas cards, then showed me the button she had made with it, then showed me the tote bag she'd printed it on. She absolutely loved the photo I made (of her dog in a Christmassy setting), but obviously was clueless about copyright infringement, didn't think about or care whether I - as a business - made sufficient sales to be able to survive to take more great photos for her in the future. Wasn't worth arguing or pursuing... Most of these cases aren't. The photographer who tries to enforce their copyright in these types of shoots just ends up looking greedy and mean... downright Grinch-like!
For that reason, part of your workflow should be to add a "signature" to your images. It should be pretty subtle (not a full size and dominant watermark), just something on the image itself, down in one corner or the other. This is enough to mark the image as professionally done and many retail printers will refuse to make copies of the image without something in writing from the photographer. The assistant will need to add this to each image before printing. In Photoshop, for example, you can prepare a few semi-transparent signatures on a layer in a file and have it open on your desktop, drag and drop onto each image. Or you could set up an action in Photoshop that adds it. Other softwares you might need to use other methods. I've also seen people actually sign their images by hand, with one of those gold pens... but be careful about smearing.
This "signature" certainly won't stop all infringements. I'm sure some people will sneak a few by the retailer, or some retailer's staff is untrained and won't stop it... Other people just used their own scanners at home. But it will help with some. An alternative might be to put a small copyright notice on the back of the print... However many inks will not dry properly on the back side of many inkjet/dye sub photo papers. Stamp pad inks often continue to smear for hours or days, if they dry at all. A way around this is to pre-print small labels to stick on the back of the print (including your URL to help people purchase more online later). However these are pretty easily removed, so might not help much preventing infringement.
A nice extra to offer are inexpensive mat/folders that you put the photos in to protect them. These are available from a number of sources in various sizes and finishes. I bought a supply in several sizes from Papermart.com and have used them at various events for years. I also add a label to the back of these, with my name, URL and phone number. Plain envelopes to fit the various sizes are a good idea, too.
Have plenty of extra ink. You can't run out. It's happened to us at events and means scrambling to find ink close by (and argument for using a very common type of printer that's easy to find supplies for)... You usually end up paying extra for it too, since there's no time to shop around an unfamiliar territory for lower prices. We were fortunate at one event that a store manager was kind enough to let us in a half hour before opening to get an ink cartridge, when someone bought sets and didn't realize they didn't include black!
Your best and most cost effective type of printer is a dye sublimation. However, to do 8x10s or 8.5x11s you are looking at a very pricey piece of equipment. You might be able to rent one in your area. But if you rent you will have to calculate your supply needs very carefully so you don't end up with a lot of unused materials that you've paid for.... Work with a store with a good return policy.
An inkjet printed can be used, but often isn't any better than most people can do themselves at home and is relatively slow. The cost of ink and paper, over the long run, is high, too.
Believe it or not, one of your biggest problems could be people bringing their own cameras and using your Santa set to take their own shots, any time you turn your back. I had that happen a lot at one event. I'd step away to swap memory cards so shots could be done printed out for waiting customers, then return to find Santa with a crowd around him and one or more people taking shots of their friends with their phones or P&S cameras they'd brought. I ended up having to wait until they finished, before I could take the next customer who was patiently and politely waiting for me. Be prepared for this! It would be best to have your set roped off and carefully control access to it. It's another situation where if you try to put your foot down and tell people to stop, the photographer ends up looking like the Grinch!
If using studio lighting, you almost need to have a radio slave or hard wired setup. People will be shooting over your shoulder and their camera's flash will set off optical slaves if that's what you are using in a multi-strobe setup. This will often screw up your shots, because only the slaved strobe(s) fired and is still recycling when you take your shot.
You should probably shoot JPEGs, or perhaps RAW + JPEG. The JPEGs are needed to speed up the process, but mean your exposures and color balance need to be pretty much on the money. Post the pictures online afterward, some people won't stick around long enough to get a print made or will want to think about their choices, or might come back and buy a different one from you to make Christmas cards or something like that. This is when it's handy to have RAW files, in case more post processing is needed.
Will you be collecting payment? Check local laws about sales tax and such. You'll need a cash box, some change and a calculator, at least. You'll have to decide whether or not to accept personal checks. Many businesses in my area no longer do. To accept payment via debit/credit/check card you'll need a full setup... talk to your bank. It ain't free! Sometimes you can get around it with PayPal or similar, if you have a wireless Internet connection.
Oh, and I charge extra for printing onsite. The reasons are mostly the cost of the assistant and the higher cost of printing supplies. Where a 5x7 costs $10 ordered online, it costs $11 onsite. An 8x10 is $20 online, $22 onsite. The print quality (inks, paper, image post-processing, working with calibrated system, etc.) are always better when handled later, online, compared to rushing to get the print out to the customer onsite. Online shoppers also get to choose from over 100 print sizes, finishes and printable products, instead of the 2 or 3 we are limited to onsite. So the customer actually ends up paying more for lower quality onsite, and has far less selection. On the other hand, the onsite buyer isn't being sharged for shipping and handling. Some people "just gotta have it now" and it's best to offer both onsite purchases as well as online sales, to be sure not to leave any money on the table, so to speak.
I'm actually set up now and will be doing Beta testing of a "Kiosk" arrangement through my printing service vendor, where people can view images on a computer onsite, place their own orders immediately and make payment with a credit card or arrange to send payment by check later... I'm not collecting any actual payment or doing any actual printing onsite. The order is uploaded securely later, along with the images files and all their shipping/billing info, and the prints are sent out to them within a few days. Much easier and less labor intensive, it's a compromise that still allows them to make that immediate purchase while they are still excited about the event, and it gives time for their images to be properly and much more professionally finished, and they have choice of all the same products I'm able to offer online. Might be the best of both worlds.... we'll see. The hangup has been that the images still need to be loaded into the computer and go through at least a minimal sorting and editing process, which still means an assistant. And, I'm no IT expert and have been struggling to get wireless image file transfer to work. (All this is just food for thought.... possible future enhancements. It's not stuff I'd recommend anyone try until they have a few similar jobs done successfully and the bugs out of the rest of their workflow.)
I'm sure there's more I'm forgetting. Frankly I try to avoid Santa gigs now. It's been a few years since I've done one. Bah! Humbug!