There's no set rule. The thing is, if this is going to be a service to you and not a nightmare to coordinate, then they have to work with you! We've done meal chains several times in my neighborhood, and we always work so that the person in need has to do practically nothing. It's perfectly fine for you to call the church (or have your husband do it - surely if you are so sick, he can take 20 minutes off of his 16 hour days and help out!).
We've used a free website called Lotsa Helping Hands many times. It's very easy to manage. You would appoint ONE person (perhaps the church person or a good friend) to coordinate. People can join the website only with that coordinator's permission. The person in need sets the schedule, lists preferences/allergies/aversions, determines when food is delivered and where (we've often put a cooler on the porch or breezeway or by the garage so you don't have to open the door), etc. We specified disposal containers so no one had to wash/return, and you can specify if you want foil pans for the oven or non-metal so they are microwaveable. You set a delivery time (we always picked something like "between 4:30 and 5:30" and indicated the location of the cooler. You can list preferences ("chicken, pork, no beef" or "no nuts, no dairy"), whatever you want. It doesn't have to be an allergy - it can just be what you prefer.
People signed up for the days they wanted, and they listed what they were bringing. That way, the next person could see that lasagna was coming on Wednesday so they wouldn't duplicate that for Friday. You can also specify kids' meals if you want to. And you can list the take-out restaurants you like so someone who doesn't cook can still order you a meal for delivery.
We also used Lotsa Helping Hands for chores like laundry and errands, carpooling and homework help, yard work and so on. It's pretty versatile and the tech support is very helpful and not overly techie at all!
If they don't use the website, you can still list the preferences with the coordinator and let her work it out with people. It does NOT help you if people just bring whatever they feel like. I'd take advantage of the offer and extend it. If you think you really need help for a week, then sign up for 3! Really! You will be way behind from being laid up and it will be a nice treat to have dinner prepared for you even if you could manage to drag yourself to the kitchen. DO let people help - just get rid of the scheduling and organizing and put that on someone else's shoulders.
I think it's okay to say you want food dropped off and you don't want people to visit/stay. The church may ask you if you want someone to stay, as there are often people who really need the company. But if you aren't up for it, then say NO! The point is for them to help you, not for you to have to extend yourself when you feel lousy.
Feel better - I hope this isn't too serious and that you will be back on your feet soon.