Combine the onion, garlic, ginger, horseradish, turmeric, and cayenne to taste in a glass jar. Add enough vinegar to completely cover all the ingredients.
Cover the jar. Vinegar erodes metal, so use a plastic, glass, or cork lid when making infused vinegars, or put a piece of waxed paper between the lid and the liquid.
Set the mixture aside to steep in a cool, dark place for 4 weeks.
Strain out the herbs using a fine-mesh strainer or fine cheesecloth or cotton muslin.
Sweeten with honey to taste, shaking or stirring well to dissolve. Store at room temperature; it will keep for up to a year.