Thickeners are sometimes very eccentric. They just need to be coddled.
For instance, xanthan will certainly work in your application, if you treat it nicely - or get the right xanthan gum. There are various grades of xanthan gum, even various grades of the food grade gum. One grade is called readily dispersible - but it is made by using formaldehyde, and restrictions on formaldehyde are currently making it difficult to find.
The problem with xanthan gum is that it disperses readily in alkaline solutions (pH ~10), but begins swelling in lower pH solutions so fast that the particles become sticky on the surface and clump together so quickly that the inner portions of the particle are then removed from easy access to water which would swell the particle.
So, one solution would be to get food grade (is that necessary?) readily dispersible xanthan gum. Another, slightly more complicated, would be to disperse the not-so-readily dispersible xanthan gum in water with a slightly alkaline pH (pH 8 will have 100 times more OH$^-$ ions than H$^+$ ions), until the mixture is smooth (and way too thick!), then dilute with your vinegar and mix thoroughly. I've done this on a large scale (1000 gallons). A little NaHCO3 or even Na2CO3 will get you the pH =8, then the vinegar will totally overcome the alkalinity, and the xanthan gum will be already swollen and thickened.
Some thickeners may work better at the low pH of vinegar, like the suggested modified celluloses such as methyl cellulose or hydroxyethyl cellulose, but making a pre-thickened mix and then diluting with vinegar might work better in all cases.
In a similar thickening situation, egg albumen coagulates in vinegar. "...by adding vinegar, we get ...increased acidity to help the egg white coagulate and form a solid white". (Ref) So don't bother trying egg white to thicken your vinegar!