Having until recently run an ion beam lab with both a High Voltage Engineering accelerator and a National Electrostatics Pelletron system, and having run a number of other similar systems over the years, I have a reasonable idea of what can be done.
Before going too far, I'd like to touch on a few factors that might not be well understood. First, an ion accelerator is not normally run like the lab-display generators, which just charge up a sphere and leak current somewhere/somehow. Instead, one wants a steady voltage on the terminal and that is only achieved through matching the current going both to the terminal and away from the terminal. At steady state (constant voltage) the two are the same. So, in the absence of an actual ion beam, the charging systems sends X amount of current up, (X-C) goes down the accelerator tube through precision resistors to set a steady voltage gradient, and C goes out as corona current, an intentional flow of charge from the terminal to the corona points. The corona current is used as the fast feedback loop on the voltage (much faster than throwing more/less charge on the belt/chain and waiting for it to be mechanically transported to the terminal). Once you add an ion beam in, you need to add more current going up to the terminal while keeping the column and corona currents the same (to keep the same potential on terminal). Should the beam current be too large a proportion of the charging current, the machine can become unstable to variation in the ion source. If you run the terminal at lower voltage, the current required is smaller (V = IR down the column). So, you might get more beam current possible at lower terminal voltages.
Second, there are two types of Van de Graaff type accelerators to consider, the single-ended systems and tandem accelerators. In a single ended machine, the ion source is housed in the terminal, making positive ions which are then accelerated down to ground. The tandem accelerator puts the terminal in the middle of the tank, and accelerates negative ions to the terminal, strips electrons in the terminal, and accelerates all the various positive charge states back to ground. So, two things with a tandem - you have two accelerating columns, so twice the column current, only one corona, and even more possible variation in beam current if the source blinks (since an ion traversing the machine goes both up and down). I will add that a positive ion source makes a lot more current than a negative ion source - it is just the nature of the beast. So, max current will be possible out of a single ended machine.
Third, modern systems like the Pelletron actually double the charge capacity of the chain by taking positive charge up to the terminal, and negative charge down from the terminal. The rubber-belted classic High Voltage Engineering machines don't do that.
So, what can a modern machine do? Well, NEC will sell you a duoplasmatron source rated at 10mA of H+ output. Since those are meant for their systems (including in-tank mounting), one can be sure they can be used at those ratings in one of their machines, provided there is enough charge available. For high current applications, NEC sells machines outfitted with up to 4 separate charging chains, so they can move a lot of charge quickly. (Now, I'm not sure I know anybody who has wanted to max out their single ended system - the real reason the source is rated at 10mA of H+ is that other ions are harder to get out of the source, so it is way over-spec'ed for H.) (I also bet that one could get more than 10mA H+ out of the source, it just might not last as long or be as steady.)
A tandem is, as noted above, trickier, since you need both high negative ion current and you worry more about beam fluctuations. I have gotten 1mA of Si- out of an NEC SNICS source and run that into a high current (4 chain) 1.7MeV tandem. I was trying to get reasonable beam current for the +9 charge state (so 17MeV Si ions). During one run, a stray lower charge state beam managed to melt a hole in the magnet chamber since it was depositing enough power in a small spot on sheet steel. For lower charge states, one would not bother to run that high an input current - it just wasn't needed. I also only ran that beam once I had lots of experience with the machine and the source and was confident it was all reliable. One flicker of the source and there would have been a big tank spark.
So - beams of up to 10mA from a single ended machine are viable, but terminal voltage, charging capacity, and machine type all play into the equation.