Many of the So Cal contests have had a People's Choice component for the past several years. It's extremely popular with the public, and I think, will ensure public support for your event in future editions. It's a win for the public, a win for the sponsors, and a win for the teams.
I don't know about your local health department (from here referred to as HD) regulations, but that's the first place to start. In California, it's enforced on a county by county level, I don't know about Kansas. People at the HD are used to dealing with restaurant chefs who already have a background with food safety, so if you don't already have your ServSafe certification, I'd advise you to get one. That way, you'll be speaking their language and they'll feel more at ease in giving you a permit. If you go in knowing nothing about the food safety laws they enforce, they won't give you the time of day. It's not their job to teach you this stuff, there are schools across the country that do. For about $125, you can take a one day class with test at the end for your food handler's cert.
Ask what requirements the HD has for a one time event where teams mostly don't have a HD department license will be selling food to the public. Requirements for a one time event are a little different from a permanent outdoor kitchen, or an outdoor kitchen that's used on a semi-regular basis. Those setups have stricter requirements for plumbing and flooring and such.
If you have a ServSafe certification and have passed your food handling manager test, this stuff should be familiar. Out here, there are some other requirement for an outdoor kitchen:
Enclosed tent for the prep area, with screens to keep out insects (pop up ten with screens are ok)
Ground covering (plastic sheeting is ok)
Three basin sink with hot water for dish washing. Santizing basin will be tested for proper strength with test strips.
Separate hand wash basin with hot water and soap
Cold food stored below 40F
Cooked food held above 141F
Food storage containers including coolers must be at least 6" off the ground.
Food handlers wearing disposable gloves
to name a few
Some contest organizers have provided a food prep tent that teams can share. At other events, each team participating in PC must have their own setups. Getting sides for your tent, a ground tarp, etc, isn't much of an expense if you think about it. The HD sends inspectors to the event and each team will be inspected. Work with your HD and ask if they can finish inspection before 11am on Saturday's crunch time, or else you'll have some pissed off teams and a pissed off inspector. Remember, the inspectors do spot inspections at restaurants unannounced, and they're accustomed to drop in during lunch service and disrupting a restaurant chef's busy time. Let them know if they pull that at a contest, they're asking for a fist fight from amateur cooks whose aren't beholden to a health department shutdown.
Once you get past this stage, make clear to your teams the HD requirements, because the first time we did PC, many of the teams weren't clear on it, and there was a significant amount of skepticism and pushback. Once we got past our first PC event, teams were very much supportive of the experience.
The way we do it, the teams are selling off the leftover contest meat, and following the usual timelines for contest turn-ins. If you make me cook meat for a separate 10am turn in, you're forcing me to cook a separate batch of meat on a different timeline than my contest meat, and I won't participate in your PC contest. KEEP THIS AS SIMPLE AS YOU CAN FOR THE COOKS.
The teams have the option to cook extra meat for PC if they have the room, but it's not required. Some contest organizers are 501(c) charities, and they get meat suppliers to donate pork butts and briskets to the event. The event organizer then gives this meat to the teams that have cooker space with the express purpose of selling it for the PC contest.
The public buys sample tickets at a desk ($10 for 5 sample tickets, and they can buy as many as they want in $10 increments). They are given one more ticket of a different color for their PC vote per each $10 buy. The tickets are sold at designated (multiple, conveniently located) tents so teams aren't touching both food and money.
The teams taking part in PC have an official event sign at their tents (WITH THE TEAM NAMES ON IT), and the public walks from team to team for their taste. It's also a chance for the team to interact with the public, and put on a show. Friendly trash talking between teams, and "selling" your food is part of the experience that the public loves. My team's won 6 PC championships, and it's also part of the fun for us too.
We use 2 oz samples of meat, served in either a plastic cup (like the type used to hold sauces at a restaurant, or a little paper "boat"). They pay for the food with those tickets, which the team collects. The team keeps count of those tickets and returns them to the organizer later to collect their money.
The money split is entirely up to you, but generally, the team earns $1, and the organizer earns $1 for each ticket. The tax implications of that money are a separate matter, and you should talk to your tax adviser about how to handle that.
Regarding the PC vote ticket: the public gives their ticket to their favorite team, who turns them in with the money tickets at the end of the day. Just make clear to the public, the teams, and all your event volunteers how both sets of tickets are going to work (where they're sold, how they're turned in, what time the PC sales end, what time PC votes end, what time the tickets have to be turned in). We have contests where nobody is told anything, and we all look like idiots because the organizer didn't share this info at the cook's meeting, or the volunteer's meeting.
We have a tradition of a Friday night cook's potluck. It's not a contest, just a good social event. I personally have too much work to do to worry about yet another competition on Friday.
Any team that puts in an honest effort at PC should make back their expenses, and that's the incentive for the teams that don't consistently finish in the top 5.