Hi. In my opinion this is not something to be too worried about and your ear should be able to heal in a few days provided no further injury occurs to the Eustachian Tube (ET).

The ET connects the back of the nose to the middle ear region (This explains why it hurts every time you burp since burping closes the ET momentarily). The ET is important for equalizing the pressure between the ear and the atmospheric environment. During the time when you vomited, the acid in your vomitus must have reached the ET opening due to the increase pressure while throwing up, leading to swelling, irritation and blockage leading to a negative pressure within the ear which pulls the ear drum inwards leading to pain every time you burp.

I would recommend that you allow the region to heal for atleast a week and avoid consumtion of alcohol, smoking or carbonated drinks during the period. YOu can expect recovery within a week.

Hope this information was helpful. Do let me know if you have any further questions.