Ok, so I didn't exactly find a recipe, but I found a few things which you could probably combine together yourself to make a real recipe. It might take some experimenting, though. First, on wikipedia, I found the traditional method of preparation:
"Shawarma is made by placing strips of meat or marinated chicken on a stick; an onion or tomato is placed at the top of the stack for flavoring. The meat is roasted slowly on all sides as the spit rotates in front of, or over, a flame for hours (see rotisserie). Traditionally a wood fire was used, now a gas flame is common. While specialty restaurants might offer two or more meat selections, some establishments have just one skewer.
After cooking, the meat is shaved off the stack with a large knife, an electric knife or a small circular saw, dropping to a circular tray below to be retrieved. Shawarma is eaten as a fast food, made up into a sandwich wrap with pita bread or rolled up in lafa (a sweet, fluffy flatbread) together with vegetables and a dressing. Vegetables found in shawarma include cucumber, onion, tomato, lettuce, eggplant, parsley, pickled turnips, pickled gherkins, cabbage, and in some countries, such as Romania, Bulgaria, Jordan, Israel, or the United Arab Emirates, french fries.
Dressings include tahini (or tahina), Amba sauce (pickled mango with Chilbeh) and hummus, flavored with vinegar and spices such as cardamom, cinnamon, and nutmeg. Chicken shawarma is served with garlic mayonnaise, toum (garlic sauce), pomegranate concentrate, or skhug (a hot chili sauce). Once the shawarma is made, it might be dipped in the fat dripping from the skewer and then briefly seared against the flame. In Israel, Syria and Lebanon, chicken shawarma are toasted after being made up, whereas those made of lamb or beef are immediately eaten.
Beef can be used for shawarma instead of lamb, and turkey is used instead of chicken. In Saudi Arabia, goat is as common as beef or lamb. Less common alternatives include fish and sausage. Some shawarma stores use hot dog buns or baguettes, but most have pita and lafa. Sometimes, beef shawarma—despite its name—contains some lamb in addition to the beef, to ensure juiciness."
Here's an actual recipe for the spices:
1/2 pound lamb meat, cut into strips
1/2 pound chicken breast, cut into strips
1/2 teaspoon ground cardamom seeds
1/2 teaspoon ground cinnamon
1/2 teaspoon ground nutmeg
1/2 teaspoon ground black pepper
1/2 teaspoon ground cayenne
1/2 teaspoon salt
3 cloves garlic, minced
1/2 onion, minced
1 bay leaf
1/2 cup lemon juice
1/2 cup red wine vinegar
6 large sections pita bread
2 tomatoes, chopped
6 slices pickled eggplant (optional)
3 serrano or jalapeno chilies, stems and seeds removed, chopped
Yoghurt to taste
Place meat in a glass bowl. In another bowl, combine the spices, salt, garlic, onion, bay leaf, lemon juice, and vinegar, stir well, and pour it over the meat. Marinate in the refrigerator for at least 2 hours.
Drain meat from the marinade and pat dry. Grill meat over a hot charcoal or gas fire until done, about 10 minutes, turning often. Divide meat into pita bread sections, add tomatoes, pickled eggplant if using, chilies, and a tablespoon or more of yoghurt.