Since there are several suggestions in replies here, I would like to tell you that I totally agree with LK Tucker. Since you don´t feel any vibration initially when braking at low speed, your rotors can´t be very much warped when cold. Thus, it is likely that they warp when heated up (during driving) from a pad lying against the rotor. And as Tucker says, this comes from bad calipers. In my experience the parts you have to buy to renovate them are very expensive, and it is more economical to buy a renovated caliper. There are xchange programs, where you pay a deposit, which you get back, when you return your used one. Try changing the most offending one (the side which is vibrating the most). Also change pads. If vibration is much less, but does not go away totally within a few 100 kms driving, then change rotor(s) as well. If you, already now, find that they are very worn, change them already now. Good preventative measure, and they are a safety part.

Remember that it is very difficult, just by looking at the caliper or moving the pistons, tell whether the return action in the piston seals is sufficient to bring them back far enough from the rotors.

This work is fairly simple, and good advise is in the FAqs and in manuals like the Haynes. And you will enjoy the feeling of fixing this yourself.