Well silly string starts out as a compressed liquid, then upon expansion and contact with air becomes solid (semi). That stuff fires about 20-30ft, so a modification of this as opposed to solenoid driven spinnerets would be most practically possible.
There's that lame, giant, takes up most of your arm toy web shooter that basically has a can of silly string strapped onto a kids arm and the palm press merely pushes down the release nozzle. But the basic theory behind it is sound.
Imagine a miniature (approx 10mm D, 40mm L) aerosol can of silly string (for arguement's sake as we've yet to have a really good idea for feasible webbing) with the release hole on the top as with many new deodorants. Just in front of this hole is a small metal plate (20mm L, 10mm W, 1mm thick) with a hole in the middle slightly smaller than the size of the plastic nozzle on the end of the aerosol (nozzle 5mm D, 5mm L; plate hole 3mm D). Set behind the metal plate is another of the same size that has a semicircular groove slightly smaller than the aerosol can (approx 8mm D). The aerosol rests in this groove as a means of fixing the can in place/loading the cartridge. The groove rests in between the end of the can and the start of the nozzle.
The grooved plate has two short, fixed arms on either side (6mm L). The first plate is attached to these arms by way of sliding up and down them (sandwich-like)
A current is sent through the grooved plate to turn it into an electromagnet due to the workings arranged just behind the aerosol can where the other fixing point for the can is. This mechanism is triggered by two micro switches in the palm of the hand - both must be pressed for power to run to the mechanism
When this happens, the first plate is snapped back to the grooved plate, thereby bringing the nozzle down with it. The pressure is released as with any other aerosol, and the fluid is fired.
Sounds dumb I know, and it might take several micro-solenoids behind the cartridge housing to illicit enough force. But it makes the issue of several container linked up, and CO2 issues a bit easier.