Electromagnetic waves do travel with the speed of light .
ie This can be derived using the 2nd and the 4th Maxwell’s equations. to get E/B = c ,
E-electric field ,
B- magnetic field
c - speed of light
Even visible light is a part of electromagnetic wave that is why it travels with the speed of light.
(but remember that electromagnetic waves travells at the speed of light only in vaccum. The speed will change in a medium accodring to the snell's law)