LabVIEW is typically not the problem with high channel count
applications. Depending upon whether or not you want to stream
continuously, your data acquisition boards or the computer backplane
are typically the areas where problems occur. That said, on a
typical modern PC, data rates of 50MBytes/sec and greater are usually
achievable on a single computer.
Don't let the fact that you have to measure current throw you. A
simple small value resistor converts you current measurement to a
voltage measurement (you will need to accurately measure its value and
pay attentions to changes with temperature). There are lots of
choices for high-channel-count voltage measurement. You may also
want to consider USB devices. They do not support as fast a
streaming rate as you will get with a plug-in, but you can daisy chain
quite a few of them.
In short, you can probably do what you want to do. Post a few
more specifics (data rates, number of channels, what each channel
measures) and we can help you more.