How do we develop concise production functions from a set of equations
It goes the other way around.
VAR's are a special group of models for time series data, developed mainly for atheoretical forecasting purposes. In a VAR approach we do not assume some structural economic model, we just want to use the past of a variable and its interelations with other variables, to form a good quality predictor (and it is for this reason that when they were first introduced and propagated by Sims, they created a lot of controversy).
In general, estimating a production function of a single firm using time-series data does not alter the econometric approach described in the answer to your other question, only here the assumption we make is that the unknown parameters refer to a single firm (and not to many firms), and so stay constant across time (and not across firms on the same time period.
A new econometric issue is that with time series data we are forced to deal with the aspect of serial correlation (in a cross-sectional setting, "serial" correlation makes no sense because we can permutate the index without consequences). Serial correlation may emerge in two ways: a) In the regressors b) in the error term.
A new economic issue in a production function setting is the intertemporal change in Total Factor Productivity.