In principle, the answer is yes, with software instruments it is feasible to (re-)set the tuning so that you can realize music with modulation that stays in just intonation across these changes. The frequencies are directly accessible in sound synthesis environments like PureData or Overtone, and even just by setting the tuning information in a set of MIDI data.
The main issue is deciding how/when/where to use which frequency for a given key on the keyboard. For standard keyboards, there is the base ambiguity between enharmonic notes: obviously between C sharp and D flat, which have different frequencies in JI, and less obviously between C double-sharp and D (and so on). In 5 limit JI there is an additional 2 fold ambiguity for a note of a given name! The D that is a third above B-flat (B-flat is 2 fifths below C) is tuned differently than the D that is two fifths above C. This issue is alluded to in @ttw and @Todd answers: for even moderately complex music, making just intonated chords would require on-the-fly decisions about how to assign frequencies to the notes. To date, no automated software system has enough "understanding" of musical context to do this at all, let alone in real time. In sequenced music, it is possible for the composer to assign these frequencies, but then they're already approaching their composition from a point of view that significantly diverges from common practice harmony.
A quick search resulted in a video that involved changing the root tone of the just intonation to the septimal seventh. I'm sure that many other people have experimented with these capabilities, but it's not something that has, in my perspective, taken off. I suspect that something like the following is going on: If I have the desire and power to reset the frequencies of all of my sounds at will, and I'm not trying to recreate conventional sounds (as Todd pointed out, synthesized sounds always sound synthesized) I can do much more novel, or extreme, things than just modulate in the conventional ways of classical music.
To summarize, yes software synths (of various sorts) in principle allow you to realize harmonically complex music in just intonation. particularly for seqeuenced music. For actual performance with a keyboard, the complexities involved in dynamically modifying the frequencies hard to address: it is too much for the performer to tweak the intonation on the fly (though this kind of keyboard might help), and automated support is not yet available. The prospects for sequenced music are more promising, but still, for conventional harmonic music the effort to benefit ratio is pretty low: you have to construct a complex system that goes beyond current standards to specific which frequencies you mean, and the problem of creating "good sounding" harmonic music is already reasonably well solved by other temperaments. It's only when you go to more extreme experimental music that exploiting this fine level of control gives you bang for the buck, but then you've moved beyond conventional harmonic practice altogether.
Note: this answer assumes that you're considering music that "sounds synthesized", getting an organic feel to the music would require better (more authentic) intrinsic sound generation capabilities, and an ability to realize expressive intonation.