In the UK, horseradish sauce is traditionally made using grated horseradish, vinegar, sugar and double cream. In the US it seems to be more often mayo, not cream. From what you say, you are looking to preserve horseradish which doesn't contain either. This essentially means using only vinegar, sugar & salt and possibly adding garlic (or spices) if you wish. The Amish one doesn't contain sugar so you could leave that out.
There is a method for making the horseradish sauce in a blender. Peel the horseradish root and cut into small cubes. Put a few cubes in the blender with the other ingredients. Start blending & gradually add more cubes. Blend until you reach the desired consistency.
Edit: Just found a recipe which uses this method and doesn't contain sugar, cream or mayo.
Homemade Preserved Horseradish Sauce - Fermenting for Foodies