This would really be a problem for someone who has to board. I not sure what to do, maybe call a meeting, and see if any ideas can be thought up. Trust is so important is such a group or boarding setting.
My horse is at a co-op barn similar to this. We have never had an issue of feed getting stollen. I put out 7 days of labeled and measured feed in ziplock baggies and SmartPaks. In this readers case since the feed stall cannot be locked up I would then keep my extra feed locked up in a garbage can or a plastic trunk. We also keep extra hay in our feed stall, I have 7 slow feed haynets set up as well.
First, call a meeting with all of the other boarders. Establish clear communication and make sure that there's not simply miscommunication going on about who owns and feeds what. If you can, assign everyone that's able different "shifts" for a week, with everyone coming out at different points throughout the day. That way, you have eyes continuously. If possible, look into splitting the cost of a few hunting cams. Those can catch the theivery in action. If the stalls cannot easily be locked, you may be able to get creative. Depending on their structure and the rules, you might be able to wrap a bicycle lock or padlock through the bars/walls/etc. If that's not allowed, or somehow physically impossible, invest in some lockable tubs. Store your grain in one, and keep it locked between uses. I'm not sure what your hay and turnout set-up is, but perhaps go in with some of the other boarders and invest in some round bales, instead of square bales. It's harder to steal those by hand! Definitely retie square bales up if you're not feeding the entire thing at once. Finally, you don't mention whether the stalls not being able to be locked is a rule, or a physical issue. If it's a rule, I'd find another barn - even if it means a longer commute! If you're not allowed to protect your expensive feed and hay, and the owner won't do anything about it, it might be time to find somewhere else.
I had this problem at our co-op barn and my friend and I had a good idea who was taking my feed. When there was an opportunity to have everyone together my friend brought this up casually, asking in front of everyone if I had any more feed go missing. The issue stopped immediately. No one was accused but it was brought out into the open and made aware to the person doing this that we knew. This was a "low stress" way to address the problem. No hard feelings were made but clearly stated that we knew about the theft.
Why can't you lock them? Maybe a chain with a padlock around the stall door (through the bars or something)? Either that or I would start bringing my feed and hay home with me every day (not easy but what else is there to do?). Or, to eliminate theft altogether, you could all agree on a feed and hay and purchase it together. That way, you are all feeding from the same pile, so to speak, so there's nothing to steal. And if someone didn't pay their part of the feed, you know who was stealing it in the first place.
I would invest in lockable containers such as trunks or even those bear proof garbage can containers for your hay. At previous barns that I have boarding at we used the bear proof containers for hay and other lockable barrels and trunks to keep mice and other critters out of our feed and hay.
I had a feed thief at a farm I used to lease. The thief leased the property over the fence and our mutual landlord didn't want to get involved, so I got chains and padlocks to put on my feed cans. I fed round bales, so that was not an issue. I also found another place to board as the neighbors were ugly in other ways. I'd agree with the folks suggesting locking the stalls. Ultimately, it'll be easy to find out who the thief is. I had my hay disappear at a self-care facility and noticed that another boarder's three bales never diminished. That and the fact that she was going around bragging about it: "She's such an airhead, she probably doesn't even notice she's missing the hay!" I confronted her with: "I'm not stupid, you're a thief. These are your options: Pay me for the hay you stole, replace it, or I call the sheriff. Everyone you insulted me to is willing to talk." I got my hay replaced a few days later. Good luck, I hope you can resolve this and not have to move.
I'm not sure why things can't be locked but what about a cheap security camera? And if most people are in agreement it seems you could work together to figure out the culprit. Even though there's not a stable nearby the thief may be from outside. How about reporting this to the police? Theft is a crime and even if they can't figure out who's responsible it may scare them into stopping.
Personally, I wouldn't put up with any kind of drama in my happy place. If you are unhappy with anything and the people in charge are unwilling to help, then it's time to move on to greener pastures before bigger issues arise.
It might be hard to figure out who the thief is. Instead, protect your own feed. See if you can install a lock on your feed stall, with your stable manager's permission, of course. If not, store your feed and hay in locked storage containers and smaller items in locked drawers. Tell other boarders to do the same. This will discourage any future thefts. Who knows, the thief may just be a hungry horse!