In a large food processor or blender combine egg product, lemon juice, mustard, and garlic; cover and process or blend until combined. With the processor or blender running, add oil in a thin, steady stream. (When necessary, stop and scrape down sides of blender.)
Add anchovies, parsley, tarragon, chives, and green onions; pulse until blended. Add the softened cream cheese and vinegar; pulse to incorporate. Cover and chill for at least 1 hour to blend flavors. Season to taste with salt.
Serve with vegetable dippers. Any leftover dip may be stored, covered, in refrigerator for up to 1 week. Makes 2-1/2 cups.