Fast-food restaurants are rushing to add meat-free burgers to their menus, hoping these higher-priced alternatives will help them capture additional traffic and dollars even as suppliers have struggled to fill all the orders.

Imitation meats made by Beyond Meat Inc. and Impossible Foods Inc. are on sale at nearly 20,000 restaurants across the U.S., according to those companies. Fifteen percent of U.S. restaurants offered meatless burgers in March, according to a Technomic Inc. study of menus from 6,000 operators, with the...