hello and welcome to HCM,
When you vomit, some food particles can easily get into the mouth and throat. Also tonsils have small and large cavities in them which we call the crypts which are notorious areas for lodging of these particles.
You must definitely visit an ENT who will have the proper instrumentation to visualize the lesion and give you a proper treatment.
Also do not try to remove that piece by yourself. Tonsils have very rich blood supply so even a small injury
can cause massive bleed.
hope this helps
Dr Sriram Nathan