Progressive waves are those waves which transmit energy from one point to another point .They have two types of waves; transverse waves and longitudinal waves. And transverse waves have further ,mechanical transverse waves( water waves) and electromagnetic transverse waves (light waves.)But longitudinal waves have only mechanical longitudinal waves(sound waves) and no electromagnetic longitudinal waves Can we conclude from this that ;progressive waves need a medium for their propagation as longitudinal waves have no electromagnetic waves?
Let's untangle the question a little bit:
Progressive waves are waves, that transmit energy from one point in space and time to another point in space and time. There are different types (longitudal/transverse), but the fundamental property, that distinguishes progressive waves from different waves, is the energy transfer.
So if the question is 'can electromagnetic waves be progressive waves', we have to check, if they can transfer energy. The answer to that question is: Yes! Even in vacuum, electromagnetic waves have an energy flux, that is described by the Poynting vector $S$
$S = E \times H$
with the usual electric field $E$ and magnetic field $H$. Mind, that this is not the magnetic induction $B$, but the two are closely related.
Even though you are right, that electromagnetic waves are purely transverse in vacuum, this does not stop the wave to carry energy with it. But in materials, there are even longitudal components to the wave.
If you want to have experimental evidence for this, there are two easy ways to check. The first is to set up a black body in vacuum and illuminate it with a focused beam of light. Over time it will get hotter, so there needs to be some energy transfer happening. The other way is on a much larger scale: our earth, as it is radiated on by the sun. Between the earth and the sun is approximately vacuum, so energy needs to be transferred through that, otherwise it would be very cold on our home planet.
In conclusion, progressive waves do not need a medium for their propagation. Electromagnetic waves in vacuum do not have a medium they exist in and still transmit energy, so they are progressive.