The issue of what constitutes a measurement in the QM sense is not exactly settled. But a measurement is not the same as an interaction. Rather, when two particles interact, what typically happens is that they become entangled
But to answer your question:
How can the electromagnetic force act on them when they then don't have a position? How can it change physical quantities that aren't defined since no measurement is taking place?
In Classical physics, of course it would not make any sense to talk about electromagnetic forces acting on particles that have no well-defined position (for that matter, it would make no sense to talk about particles having no well-defined position). But in QM, interactions take place, fundamentally, not with particles but with wave functions. That is, while in Classical physics Newton's law F = ma tells you how the position of an object changes with time, given certain forces acting on it, in QM Schroedinger's equation tells you how the wave function changes with time, given a certain 'Hamiltonian' acting on it. Physical properties like position are typically not well-defined for a wave function, but if you know the wave function, you can calculate the probability that you'll find the particle in a given position if you make a measurement.