Our school lunch program is great and the staff are very supportive of using food from our school garden. So we're very lucky in that regard.
Does you school offer a salad bar. We have one every day and much of our fresh produce goes on this. Its a nice way to use garden produce that doesn't require a lot of planning ahead of time. When we have fresh lettuce and greens it supplements the purchased stuff (our garden isn't large
enough to supply the entire need). Same with other veggies. There is standard store bought stuff on the bar but also garden produce as it is available.
Have a special event for garden produce.
We always have a garden stew day in the fall. We harvest our potatoes, onions, carrots,etc, We spend a week or two harvesting everything with the students and then it all goes to the kitchen. The staff puts a garden stew lunch on the menu way in advance. They then have time to look over what the garden produced and purchase whatever additional supplies are needed to round out the stew. Then one the scheduled day we have a small harvest celebration, do a bit of stone soup type lessons and storytelling with the students, and they have garden stew for lunch. This works nicely because the kitchen can plan and put a stew on the menu well in advance. The produce is mainly storage type items so they can be harvest and kept for a few weeks, which gives the kitchen staff time to look over what is available and plan whatever else they need. Plus its a fun event day for the students and the teachers can choose to participate with harvest type activities and lessons but they are not required too if their schedules don't allow for it. The flexibility and choice to participate to varying degrees is really key to getting everyone on board.
One of the main hurtles to using garden produce in the lunchroom is, as you mentioned, the need to have menus and food orders prepared well in advance. We've fond the best way to deal with this is to use the garden produce as a supplement so that if it is available it adds to the dish but the days menu can still be prepared with purchased ingredients if the garden harvest fails, is smaller than expected, or ripens later than expected, etc.
The other option is to have the students eat the produce right out of the garden. My students are always begging to eat the produce when we're out there. I try to plant plenty of items the kids can just harvest and pop into their mouths. For us that is snap peas and snow peas, carrots, radishes, cherry tomatoes, raspberries, strawberries, blueberries. A hungry classroom an strip a row of peas in record time. When we do
root veggies I sit out several bowls of
water and veggie brushes and the students wash their harvest before they eat it. Another easy and popular garden snack is lettuce wraps. Each student gets a large lettuce leaf and then get to pick a few other veggies (peas, carrots, tomatoes, broccoli, or whatever is ripe). They use the lettuce like a burrito wrap to wrap up their veggies and eat as roll. Usually I give them some dressing to dip it in but not always.
So my advice is basically to plan your garden around things that can be eaten raw as finger foods with little to no prep needed and/or storage type produce that can be harvested and kept long enough for kitchen staff to make a plan to utilize it.