Can the Lorentz force expression be derived from Maxwell's equations?

Maxwell's equations do not contain any information about the effect of fields on charges. One can imagine an alternate universe where electric and magnetic fields create no forces on any charges, yet Maxwell's equations still hold. ($ \vec{E} $ and $ \vec{B} $ would be unobservable and totally pointless to calculate in this universe, but you could still calculate them!) So you can't derive the Lorentz force law from Maxwell's equations alone. It is a separate law.

However...

• Some people count a broad version of "Faraday's law" as part of "Maxwell's equations". The broad version of Faraday's law is "EMF = derivative of flux" (as opposed to the narrow version $ \nabla\times\vec E = -\partial_t \vec B $). EMF is defined as the energy gain of charges traveling through a circuit, so this law gives information about forces on charges, and I think you can derive the Lorentz force starting from here. (By comparison, $ \nabla\times\vec E = -\partial_t \vec B $ talks about electric and magnetic fields, but doesn't explicitly say how or whether those fields affect charges.)

• Some people take the Lorentz force law to be essentially the definition of electric and magnetic fields, in which case it's part of the foundation on which Maxwell's equations are built.

• If you assume the electric force part of the Lorentz force law ($ \vec F = q \vec E $), AND you assume special relativity, you can derive the magnetic force part ($ \vec F = q \vec v \times \vec B $) from Maxwell's equations, because an electric force in one frame is magnetic in other frames. The reverse is also true: If you assume the magnetic force formula and you assume special relativity, then you can derive the electric force formula.

• If you assume the formulas for the energy and/or momentum of electromagnetic fields, then conservation of energy and/or momentum implies that the fields have to generate forces on charges, and presumably you can derive the exact Lorentz force law.

I haven't seen this mentioned in the answers so I thought I should at least mention it. If you take the perspective that Maxwell's equations are the equations describing a $U(1)$ gauge field, then minimal coupling (which is, in a sense, the only gauge invariant way of coupling matter to a gauge field) ensures than any charged particle obeys the Lorentz force law, with the only freedom being the value $e$ of its charge. So while Maxwell's equations themselves, without some additional assumptions, may not necessarily imply the Lorentz force law, $U(1)$ gauge invariance does imply the Lorentz force law. In fact, if you take $U(1)$ gauge invariance as being the fundamental starting point, then it implies both Maxwell's equations and the Lorentz force law. Again, this is a matter of perspective, so I am not disagreeing with the other answers, but I think that this is the modern point of view.

Yes, the Lorentz force law can be derived from Maxwell's equations (up to a multiplicative constant), with only a few assumptions about what it means to talk about a field theory.

If we start from Maxwell's equations in a vacuum, we observe that they are Lorentz invariant. Therefore we expect that any force law had better be Lorentz invariant. If you like, you can add this as an explicit assumption.

Applying Noether's theorem for time-translation symmetry, we get an energy conservation law for an energy whose density is $u=(1/8\pi)(\textbf{E}^2+\textbf{B}^2)$. The factor of $1/8\pi$ is arbitrary and not specified by Noether's theorem. There is also nonuniqueness in the sense that you can add certain kinds of terms to this expression involving things like second derivatives of the fields, but I don't think those terms have any effect on the following argument, because the argument will depend only on the integral of $u$, not on its local density, and the added terms only give surface terms in the integral, and those vanish. This ambiguity is discussed in the Feynman lectures, section II-27-4.

Now add the source terms to Maxwell's equations. Consider two sheets of charge $\pm Q$ in the form of a parallel-plate capacitor with a small enough gap so that the interior field is nearly uniform. The energy $U=\int u dV$ is finite and calculable from the geometry. If we move one sheet closer to the other by $dx$, the energy in the electric field changes by $dU$. The total force between the sheets is $F_{total}=dU/dx$, which we can also calculate.

Now when we talk about a field theory, we assume that it's local in some sense. For this reason, the force acting on a small piece of charge $q$ in our capacitor can only depend on the field at that point, not on the field elsewhere. But the field has no transverse variation, so given $\textbf{F}_{total}$, we can infer the contribution $\textbf{F}$ from the force acting on $q$. The field is actually discontinuous in our example, but one can deal with that issue, which produces a factor of 2. The result of this example is $\textbf{F}=q\textbf{E}$, and the only possible wiggle room is that we could have chosen a different constant of proportionality in our definition of $u$. In other words, we could have changed the conversion factor between electromagnetic energies and mechanical energies, but we had no other freedom here. We could have chosen this conversion factor such that $\textbf{F}$ would vanish identically, but then electromagnetic fields would be undetectable with material devices, so this possibility is not very interesting.

Once the electrical part of the Lorentz force law is established, the full Lorentz force law follows from Lorentz invariance.