Explore the main web site, not just the forums, and you'll find the answers to your questions. Use the Search feature in the green bar at the top of the page whenever possible.
As for the bread and other packaged products, yes, there are a lot of brands of bread that meet with the program's guidelines. Go to the far left drop-down menu in the green bar, the one that says Education. In the menu that appears, choose Recipes. In the menu that appears next, you'll see a link Canned & Packaged Foods. Go there and all the brands will pop up. But always be sure to read the labels at the store, because companies are always changing their recipes, and what may have been safe and oil-free when that list was made may not be now. There are also other brands that *are* allowed but aren't on the list because they just didn't exist when the list was written up.
Also, just because one type of a product from the company, bread, for example, is safe doesn't mean all breads from that company are. Again, to use bread, Dave's Killer Bread is a brand that isn't on the list here, but has some approved types we can have. Good Seed is one. But if you check the labels, you'll find that some of the other breads from that company have oil in them, so would *not* be allowed.
As for condiments, you'll find them on the lists on the same page. Again, read labels.
Now, to save some time, I'll answer the vegan butter and mayo question right here - No, there is NO vegan butter or mayo in stores you can use. All of them are made with oil, and oil is the #1 food-like product we do NOT eat on the McDougall food plan. But if you go to the recipe section and do a search, you'll find some recipes to make approved versions of them. Now, don't expect them to taste remotely like real or vegan versions of the products, because those are not loaded with fat and salt and these aren't, but eventually your tastes will change and you'll come to accept them as decent substitutes.
Have fun exploring, not just the main web site but this whole way of eating.
You'll be glad you took the time.