How can the position of charge be determined without EM waves?  The EM waves produced by accelerating charges can perhaps be used to locate the position of the charge, as usually done with the de Broglie waves. The stronger the oscillations of the electric and magnetic fields in space, the higher the probability of finding the charge. 
But what if the charge moves with uniform velocity? 
Without acceleration, no electric and magnetic fields will be created. 
How can the position of that charge be determined without EM waves? 
 A: 
How can the position of that charge be determined without EM waves

Accelerated charges and Cerenkof radiation can be useful in experiments, but if one really wants the location of charges one needs detectors, i.e. consecutive small interactions. These are mainly electromagnetic scatterings but so soft that they are included in the measurement error. To get an idea here is the simplest detector making the track of particles visible, a bubble chamber.


This picture shows how the observed ‘vee’ could be lined back up to show that it had come from a previous ‘kink’. This helps to identify the neutral particle produced and adds to our knowledge of the particles produced at the initial interaction.

The tiny bubbles showing up in the photo are where the charged particles passed.
The detectors at LHC work with the same logic for tracking, different media.
A: In your first sentence you mentioned charges. Electrons, protons as well as their antiparticles have permanent an electric field. This field seems not to be only an induced one during any measurement. The permanent state of the electric field of charges is a postulate because without any interaction a measurement is not possible (see the comments to your question). Anyway, we believe that a charge has a permanent electric field. All the mentioned particles and neutrons too have a magnetic dipole moment and this is also a field, it is the magnetic field.
Reading your question careful it seems you want to know, why a charge in rest, which do not produce any electromagnetic field, can be observed. Your presumption is wrong. As I show above, particles are surrounded be their permanent existing fields. The electric field you are able to measure and the magnetic field of this particles too.
A EM field is something different. First at all I prefer to talk about EM radiation as the less specific case. You are right, if you say that charges (as well as neutrons) emit EM radiation under acceleration. Bouncing electrons in a wire forwards and backwards in a bulb, one get EM radiation in the infrared and in the visible part of the EM spectrum. Braking fast electrons on metallic surfaces could let to X-Ray's or gamma radiation.
Conclusions: Even in rest, particles have a magnetic and an electric field and could be detected by measuring this fields. Accelerated particles emit electromagnetic radiation and this radiation could be measured without disturbing the particle. But don't forget that the acceleration of a particle is a disturbance of this particle too. So anyway the particle which properties you will measure, is under the influence of a field. 
