Will emerging markets' current economic crisis lead to a fully fledged currency crisis, which in turn triggers an out-of-control inflation (hyperinflation) in EM countries?
No. With one possible exception: Argentina.
Once you have a severe current account crisis, the options to overcome it are:
1. A draconian and swift contraction in domestic demand via a brutal fiscal adjustment accompanied by a currency devaluation
2. A more moderate multi-period contraction in domestic demand via a moderate multi-period fiscal adjustment accompanied by a softer currency devaluation. This requires external support (i.e. IMF) to finance the (shrinking) current account deficit during the period of adjustment
3. A miraculous swing of the current account deficit into surplus as a result of a rise in exports
Given that in the Argentinean case (1) will not happen for internal political reasons and (2) is out of discussion as a result of Argentine's 2001 default and the country being at odds with all major international financial institutions, either an economic miracle happens or a fully fledged currency crisis and inflation spiral will ensue at some point in 2014.
To fully understand why, you don't need me. All you need to do is to read the 1990 classic paper about "macroeconomic populism" from the great Rudiger Dornbusch