Good idea! Do you know people with this condition who would want to exchange about it?
The main issue I see though is that if you're going to include processed foods, things will get complicated.
The names for various ingredients change in some countries ie: Sorbitol is also known as glucitol, D-Sorbitol, Sorbogem and Sorbo .
I use this example because Sorbitol is a sugar alcohol that affect FODMAP types.
You might do well having a chat with the Monash University, they have done a lot of work on this and have even published an app for people to check and see what sensitivities particular foods might have.
That database might be crucial for your site.