Why electron/ion beams are not prefered over lasers as driver for inertial confinement fusion? Most of hi-end ICF research facilities, and proposals for future powerplant or space-ship propulsion use lasers as driver. 
Problems with lasers


*

*intense lasers
needed
(e.g. NdYAG 3rd harmonic conversion on nonlinear crystal) have energy
conversion efficiency just few percent (2-5% ?). The whole business
of nuclear fusion power is about getting net-gain of energy, so
loosing 95% right at the beginning seems very stupid. 

*They are also very heavy, since active medium (Nd ions) are sparsely
diluted in heavy inert class/crystal matrix. On top of that there is
lot of cooling equipment etc.
Advantages of electron/ion beams?


*

*On the other hand electron/ion accelerators can easily achieve efficiency 80-90%. 

*Also very powerful electron beams can be made quite compact and lightweight (unlike lasers), which is very important for spacecraft propulsion. 

*To cause ablation of target or X-ray generation from walls of  hohlraum you do not need very fast particles (like hi-end accelerators in CERN) but you are fine with few keV which almost industry-grade machines can produce.


So what's the problem of electron/ion beams?


*

*One problem with e-beam drivers I read about, is that target becomes
negatively charged and e-beam is deflected out or dispersed. But you
can easily combine two beams (positive ions and electrons) to avoid
that.

*An other problem on Earth is that electron/ion beam systems must operate in vacuum, while lasers can operate in air. Maybe in future applications in spacecraft propulsion the advantage reverse. 

 A: To achieve ignition in inertial confinement fusion the driver beam (independent of its nature) must deliver about $1\,\text{MJ}$ of energy to a target size of about a millimeter in less than $10\,\text{ns}$ time interval (see Lawson criterion). This corresponds to power of more than $100\,\text{TW}$. For an electron beam the space charge effects would prevent focusing of such a beam. 
For example, assuming electron energy $E=50\,\text{keV}$ ($v\approx 0.44\,c$), total charge would be $Q=e\cdot 1\,\text{MJ}/E\approx -20\,\text{C}$. Twenty Coulombs is an enormous charge, and for a $10\,\text{ns}$ time interval it must be spread along the bunch of about $10\,\text{ns}\cdot 0.44\, c \approx 1.3\,\text{m}$  length. The electrostatic self-energy of such beam would be several orders of magnitude more than kinetic energy of its electrons. Which just means that it would be impossible to create/focus such a beam.
Ions and especially heavy ions have much lower charge to mass ratios, meaning that space charge effects for them would be significantly reduced, making the scheme viable. Indeed, heavy ion driven inertial fusion (HIDIF) or simply heavy ion fusion (HIF) is a promising candidate for a viable fusion power technology. Such power plant would require a large heavy ion accelerator with total beam power much greater than any of currently operating accelerators (although the energy per ion would be much lower than current state of the art). So, while current accelerator technology is definitely relevant, the high beam currents required would no doubt present a lot of new challenges. And the size and cost of such accelerator would be comparable with today's larger high energy facilities.
So your notion of  quite compact and lightweight (unlike lasers) is definitely wrong (at least for today accelerator technology).
For a recent review of HIF have a look at:


*

*Hofmann, I. (2018). Review of accelerator driven heavy ion nuclear fusion, Matter and Radiation at Extremes, Vol. 3, Issue 1, 2018, pp. 1–11,
open access web.

A: Electron beams cause charging and suffer from beam divergence because of Coulomb repulsion. High intensity short pulses, such as needed for confinement fusion, should be hard to achieve with electrons.
A: A good reference that describes how ion beams are neutralized, and even addresses the use of neutralized ion beams for fusion, is this.

Recent neutralization studies have concentrated on intense ion beam transport to small inertial fusion targets. In Section 5.4, we saw that space-charge forces interfere with focusing. In this section, we shall study processes that limit focusing of neutralized beams in vacuum. Although the focal spot size for a neutralized beam is smaller than that for a bare beam, we shall see that collective effects may present limitations for some applications.

The author goes on to illustrate the challenge of obtaining a high-power neutralized ion beam with a sufficiently small focal spot.  One of the key issues is the fact that electrons in the beam are heated to enormously high temperatures at the target, which feeds back on the incoming beam, causing it to spread.  However, he doesn't say it's impossible; he just makes it clear that there are big technical challenges.
