You may be surprised. About 20% of Germans are vegetarian or vegan. Every restaurant will have both options. Have yet to go in one that doesn't. You will find veggie burgers and vegan burgers everywhere. Mushroom dishes are popular, and all kinds of cheese dishes. Go for the Käse spätzle which is the German version of mac & cheese.
Butter is a no-no if you are vegan, but frying in butter is something you only use if you are making schnitzels, due to the expense. For other frying, restaurants use vegetable shortening like any place in the US would. They aren't using lard or beef fat. Asian will use peanut oil.
Yormas are just in train stations and only have sandwiches, fruits, drinks, and baked goods. Backwerk are cheap, mass produced bakeries. Frittenwerk are serving up Canadian Poutine, but you can also get the fries with guacamole, etc. We found them to be a poor excuse for a decent meal and won't go there again after trying them twice. Dean & David are a bit pricey, but their salads, etc. are decent. Hans im Gluck has pretty good food and excellent fries including sweet potato fries. They have an extra menu on the table that lists every ingredient in every dish. Great for those with any kind of allergy.
Seriously, I wouldn't worry about trying to find a special place to go. Read the menu outside of any restaurant. Vegetarian and Vegan will have a separate listing.
Alternative - use the vegetarian filter on Trip Advisor on their restaurant listings. You may be surprised at all of the vegetarian and vegan places there are in both cities.