I would suggest the Dropbox Datastore API. You can use the ParcelKit wrapper which allows you to use Core Data. This gives you everything you need: offline use, Core Data, and a Javascript API for your web component.
There is also Wasabi Sync, which is Core Data-native, and has a REST API for web use.
If you can drop the requirement for web, there are solutions like Ensembles and TICDS, which work with multiple backends (e.g. iCloud, Dropbox). (Disclosure: I develop Ensembles)