There are several mammals that don't have vocal cords. These animals tend to rely on other means of communication like body language and pheromones. Some examples of these mammals are rabbits, giraffes, and dolphins.
While dolphins are often seen as underwater, they also use sound to communicate. They don't have vocal cords but they do have a structure in their mouth called the phonic lips. This structure vibrates when dolphins blow air through it and creates sounds that echo off of nearby objects. Dolphins use this echoed sound to navigate, hunt, and identify objects in their environment.
The lyrebird is another mammal that doesn't have vocal cords but is able to produce an incredible repertoire of sounds. Lyrebirds mimic the sounds of other birds and can even reproduce the sound of a chainsaw, a toy gun, or a camera. This impressive mimicry is used to attract a mate or mark territory.
Alligators can also produce an incredible variety of sounds without the need for vocal cords. While their vocalizations aren't as sophisticated as the lyrebird, alligators can generate many different tones and pitches using the flexing of muscles in their larynxes.
Other examples of mammals that don't have vocal cords are giraffes and rabbits. These animals do make some mechanical sounds such as squeaking, chewing and clicking when they eat, but they don't make any sound through their mouths. This is because giraffes have very long necks and their throats do not develop as other mammals do. Rabbits do not have vocal cords either but they can make a loud groan or yelp when stressed.