Yes, there is some alcohol in vinegar.
Estimates of Ethanol Exposure in Children from Food not Labeled as Alcohol-Containing, Gorgus et al., Journal of Analytical Toxicology 2016 discusses a range of sources of alcohol exposure, some of which might be surprising.
Among them is white wine vinegar, which is quoted as having 2.6 g/l. That's around 0.25% ABV (vinegar is mostly water, and acetic acid has a similar density to water, so close enough). I can see no reason for other real (brewed) vinegar to have appreciably less. Consuming a unit/standard drink of alcohol would mean drinking large quantities of pure vinegar (a UK unit is 8 g of pure ethanol, so that would mean drinking about 3 litres of vinegar), so for the majority of people this wouldn't be an issue; indeed the human body produces several grams per day (Wikipedia link, but see the cited source for this claim for a full review).
However if someone is trying to completely avoid consuming any alcohol, vinegar would seem to be off limits. There are alternatives, such as "non-brewed condiment" a malt vinegar substitute made from industrially produced acetic acid, water, and flavours. Similar products may legally be allowed to be sold as vinegar in some countries - they should be cheap, and include "acetic acid" on the list of ingredients (see Wikipedia's article for links to other languages).
It appears to be possible for consumers to buy food-grade acetic acid at high concentrations. This would need to be diluted significantly. One brand I found was described as 70% acetic acid; as common vinegar is no more than about 5%, you'd need something like 1 part of that 70% product to 12 parts water (or fruit juice, or a mixture of the two) and undiluted it should be handled as a nasty chemical rather than as a food (protect eyes and skin, avoid inhaling fumes).
Industrial production is done directly, without the alcohol intermediate step, so can be assumed to be alcohol-free and would also allow some distance from the alcohol industry, if that's related to the motivation.
Returning to the paper I linked at the start, some other foods may be surprisingly high in alcohol. Some breads, for example have over 1% by weight (though others have no detectable alcohol). Very ripe fruit has some, and fruit juices contain tiny quantities, with the highest figure detected in the juices around 1/3 of that in vinegar, at less than 0.1%, though of course fruit juices are often consumed in larger quantities than vinegar. Perhaps the natural level found in fruit juices could be regarded as a practical threshold to consider functionally zero, barring any medical reason to go still further.