If the equations of motion of the vibrating system are equivalent to real and symmetric mass and stiffness terms, the normal modes will be real vectors, which means that all parts of the system move in the same phase.
That excludes travelling waves, where there is a phase difference between points in the direction of travel of the wave.
There is a special case, when two (or more) vibration modes have identical frequencies. In that situation, a combination of the different mode shapes with different phases may "look like" a travelling wave. However this may only be a theoretical possibility, because the tolerances in a real-life structures often separate the two theoretically-identical frequencies.
However there are mechanical systems which do have "travelling" normal vibration modes. A simple example is a gyroscope, where the vibration modes include precession and nutation.
In general, the equations of motion of a system rotating with constant angular velocity will include Coriolis terms, if it is modelled in a rotating coordinate system fixed to the undeformed shape of the body. The equations of motion are then Hermitian matrices rather than real symmetric matrices. The eigenvalues (natural frequencies) are still real, but the mode shapes are now complex vectors which can be interpreted as travelling waves.
In general, the speed at which the "mode shape" rotates around the object is different from the rotation speed of the object itself. If the two speeds coincide for some particular rotation speeds, that can have severe consequences for the design of real rotating machinery - for example the so-called "critical speeds" of rotating shafts.