Until a couple of years ago, there was an easy, automatic answer for the entry level -- get a couple of Epson R800s with outboard bulk inking systems, use roll paper and go to town. Unfortunately, the R800 is with us no more, and while you can do the same job with larger inkjet printers, transportation, setup and tear-down, consumables management and so forth make it altogether impractical for more than a makeshift solution.
There is a cost of doing business, and making on-site printing manageable and profitable comes with a little bit of up-front pain. The weapons of choice these days are dye-sub printers that will run you $2-4K (depending on the maximum print size you need). As business expenses go, it's something that amortizes quickly -- and since the profit is in the print, not the image, it doesn't make sense to be penny wise and pound foolish here.
The brands I keep hearing songs of praise for are the Sony SnapLab series and Mitsubishi. Both appear to be robust and bullet-proof (the Sonys are designed for retail kiosk use as well as event photography). Both use system consumables -- if you have paper, you also have ink. Yes, the consumables seem expensive up front, but when you consider that you've already sold the print before you print it, it's really not so bad. They're monolithic -- you just plug them in and turn them on, and they're ready to go. (Inkjets that see a lot of banging around often need to go through a head-cleaning/ink charge cycle before they're ready to use. Even lasers may need to have the toner carts jiggled so the toner level is, well, level.). And they're fast (you can get a 4x6 in under 20 seconds).
It's probably more up-front expense than you were hoping to deal with, but a good printer will pay for itself in only a very few events, and the consumables pay for themselves as you go.