Does the lorentz force law only consider relative velocity? A magnet moving through a fixed solenoid will produce a force on the electrons, creating a current. However, the solenoid has 0 velocity, so the Lorentz force law doesn't work.
My question is, for every electromagnetic reaction, will the lorentz force law work if and only if the relative velocity between the particles in question is considered?
 A: The Lorentz force consists of two terms: $$\mathbf{F}=\it{q} \mathbf{E} \ + \ \it {q} \ \mathbf{v} \times \mathbf{B}.$$When you push the magnet through the fixed solenoid the force the electrons in the solenoid experience is due to the first term, the electric field term. The electric field can be calculated from the Maxwell equation $$ \text {curl} \mathbf{E} = -\frac{\text{d} \mathbf{B}}{\text{d} t}. $$ (We apply this over the inner cross-section of the solenoid, over which $\bf{B}$ is changing.)
When we keep the magnet still and move the solenoid towards it, the electrons in the solenoid experience forces due to the second term, the "v" being the solenoid's velocity in the frame of reference in which we are working.
Provided that the relative velocity is much less than the speed of light, we get the same size of force whether we work in the frame of reference in which the solenoid is stationary, and the force is due to an $\bf{E}$ field or the frame in which the magnet is stationary and the force is due to a $\bf{B}$ field. This comes about because both sorts of field are really parts of a single electromagnetic field, the $\bf{E}$ and $\bf{B}$ components of which inter-transform in a special way when we change frames of reference.
I gave a fuller treatment of essentially the same issue, without the restriction that relative velocity << c, as an answer to "How to work out the induced electric field of a magnet moving through solenoid". 
A: So the Maxwell equations are in SI units
$$\begin{array}{rllrl}\nabla\cdot E &=~~ \rho/\epsilon_0&~~~&\nabla\times E &=~~ -\dot B\\
\nabla\cdot B &=~~ 0&~~~&\nabla\times B &=~~ \mu_0 J + \mu_0 \epsilon_0 \dot E.\end{array}$$These say that an electric field can either be created by coming out of a fixed charge (top left, Coulomb's law) or else by curling around a changing magnetic field (top right, Faraday's law). Your moving magnet must be described as a changing magnetic field in the reference frame where it is moving, so an electric field curls around this change and drives the electrons in their loops: the Lorentz force works just fine. Or in the frame where the coil moves towards a stationary magnet, sure $E$ and $\dot B$ are both zero, but $v$ and $B$ are not zero and the electrons move about the solenoid that way.
Now Einstein happened to be bugged by the fact that you can work out both situations explicitly (solenoid comes towards fixed magnet with speed $v$, magnet comes towards fixed solenoid with speed $v$) and even though the machinery is very different, the answers that electromagnetism gives you are the same. In fact it bugged him so much that he gave this equivalence a name, the "principle of relativity," and endeavored to see how it came from the underlying math. This was nicely already done for him by a man named Lorentz, who had worked under the assumption-of-the-day that these equations were only valid in one special reference frame, the reference frame of the "luminiferous ether." Lorentz was both very practical, taking the approach of "let's do a coordinate transform to the ether's frame of reference, do the calculations, and then coordinate-transform back," but also aware of what the equations were saying. So for example Lorentz realized that the equations demanded a length contraction which would also contract us, since we're made of ordinary electromagnetic matter. He even played around with the idea that this thing really exists but we can't measure it because our rulers must be likewise contracted.
Einstein took these equations very seriously and argued that they described reality. In the modern language I would say, "whenever you accelerate by some small speed $v\ll c$, you do not just need to modify the space coordinates of everything $x \mapsto x - v~t,$ you also need to modify the time coordinates of everything $t \mapsto t - v~x/c^2.$" Of course this needs to be carefully extended to higher speed differences via matrix exponentiation and limits; but it is the core effect of relativity which generates all of the other effects: when you start moving past a grid of clocks which are all "in sync" you start to see them all as "out-of-sync" by slight, almost imperceptible amounts. So for example if the clocks were on the ground and you were flying in a typical consumer airliner at a typical cruising altitude (~900 km/hr, can see things ~300+ km away on the ground) you would have to be able to measure the desynchronization between the furthest away clocks you can see at a resolution of nanoseconds or better in order to see the effect. But when you compound the effects over and over you are indeed forced into these length contraction and time dilation effects.
Anyway, if you use the "correct" coordinate transformation then yes, the equations are invariant; some of the electromagnetic field will change from being electric to being magnetic and vice versa in your coordinates, and you will get the same result even though you are moving slower/faster through space and therefore experiencing less/more magnetic force.
