Can one forecast $\Delta Unemployment$ and $\Delta Inflation$ using a VAR model based on Okun's law?
Well you can essentially forecast anything using a VAR/VECM framework.
If you're wondering if such estimates will be accurate the answer really depends on the region, as the Okun Coefficent differs depending on where you are. (much like the law of gravity on earth).
a working paper released by the IMF titled Do Forecasters Beleive in Okun's Law? its conclusions were rather in favor for the use of Okun's law in forcasting.
Our results show that professional forecasters believe in Okun’s Law to a degree merited by how well it holds in the data. For all nine countries we study, the relationship between forecasts of changes in unemployment and changes in real GDP growth is negative, which is consistent with Okun’s Law....
Its generally more accurate for short run forecasts then it is for long run. However the same counter-cyclical relationship is maintained between Real GDP growth rate and Unemployment Rate.
the correlation between output and unemployment forecast revisions is stronger in year-ahead forecasts than in current-year forecasts; this finding leads us to conjecture that forecasters’ reliance on Okun’s Law is stronger in the early stages of the forecast horizon, when information about the likely departures from Okun’s Law is scarce.