Does oak aged wine contain vanillin?
Oak aging can indeed produce real vanillin, whether in wines or other liquids (spirits, beer, even vinegars can all be oak-aged)  among thousands of other flavorful compounds. Somewhat surprisingly, it seems that vanillin is commonly synthesized for use in artificial vanilla extracts from lignin, a fibrous compound that serves to strengthen the cell walls in wood, or from guaiacol, an oil derived in turn from lignin.

However, this doesn't imply that you will necessarily have an allergic reaction to vanillin based on the information you've provided. Vanillin is a flavorful aromatic compound, and not a protein (which typically have much larger, more complex molecular structures). "Real" vanilla extracted from the eponymous bean gets much of its flavor and aroma from vanillin, but it contains much else besides. If you're definitely allergic to the proteins, those would be present in extract from actual vanilla beans, but not in vanillin produced from other sources such as wood, and therefore not present in oak-aged wines.

The caveat, of course, is that I'm not a doctor, an allergy specialist, or anything remotely close. You should confirm this with them before risking discomfort or bodily harm.