I'm not going to give you a recipe, I'm going to give you some guidelines. Because soup can be made with anything (do you remember the Stone Soup story from when you were a child?).
Soup starts with liquid. Add meat, vegetables, and seasonings. Cook until done.
Now, specifics.
Liquid can be just about anything, but start with water. Broth can be added at the beginning (beef broth or vegetable broth if you can't use chicken broth). Milk, cream, or (in a few instances) wine should be added near the end of the cooking process. Milk or cream would normally be added after the food has been removed from the heat; wine should cook for a little bit to cook off the alcohol.
Meat: red meat is normally cut into small pieces and browned before adding to the soup. Fish or other seafood, or ham, would be added without browning. Most fish and seafood doesn't need to cook very long (clam necks being an exception, as they are tough and need long simmering). Red meat or ham should be simmered -- your slow cooker will do a good job with this.
Vegetables can include anything you like or have on hand. Potatoes are frequently used, but other starchy foods added to soups include legumes, rice, noodles, and sweet potatoes. Cut the veggies into bite-sized pieces before adding to the soup. First add the ones that take the longest to cook, such as carrots. Then add the ones that cook more rapidly, such as, for example, beets or broccoli or kale. If you use rice or noodles, don't simmer them all day or they'll fall apart; instead, add them about half an hour before serving time.
You need something for what the gourmet cooks call 'umami' -- good flavor. Meat, including bacon; mushrooms; meat broth; milk/cream/sour cream/yogurt (the last two may be added as a spoonful on top of the bowl just before it's served); cheese....these are all used for that good flavor.
Seasonings: Most shouldn't be added right at the beginning if you are going to cook the soup all day in the slow-cooker. There are a lot of things you can use instead of onions or garlic, but you'll have to choose what you like or have on hand. Celery is often used. Bay leaves (remember to remove them before serving). Sage, thyme, oregano, tarragon, rosemary, parsley, basil, salt and pepper, bell peppers and hot peppers....there are more -- lots more!
Now, suggestions. Look for a borscht recipe (Russian peasant winter soup) if you like beets and cabbage and potatoes with a little meat. Look for a chowder recipe if you like seafood/fish. Look for chili recipes if you like Mexican style (with or without beans -- you can leave out onions and garlic). There's bean soup; beef and vegetable soup; sweet potato soup....You can use recipes that call for your forbidden ingredients, just substitute. Use beef broth and red meat (or fish) instead of the chicken. Use one of the seasonings I mentioned above instead of onions or garlic. In our house, soup is generally a 'clean out the frig' day and the ingredients vary depending on what we have on hand. You probably don't need more than a quart of liquid to get started with, a pound of meat chopped small, and three or four pounds of vegetables, along with maybe one pound of rice or noodles if you use those, or a potato per person. Test your seasonings before serving and add your salt then.
Test ahead of time, if you are really worried about it, but soup generally comes out just fine.
Kathleen