Combine the mustard, horseradish, yellow onions, green onions, celery, sugar, paprika, Worcestershire, and garlic in the bowl of a standing mixer, and mix on medium speed until well blended. With the machine running, slowly add the oil in a steady stream and continue to mix until the mixture is thick and emulsified. Slowly add the vinegar in a steady stream and mix well to blend.
Transfer the sauce to an airtight container and refrigerate until well chilled before using, at least 2 hours. (Stored in an airtight container, this sauce will keep refrigerated for up to 5 days.)