If I have a lot of data points and number of different dependent variables, can I use central limit theorem to assume data is multivariate normal and compute my VAR? Is this the appropriate use of central limit theorem for VAR calculation?
There are several methods to compute VAR: i) historical, ii) variance-covariance, and iii) monte carlo. iv) copula techniques. I assume you are asking about approach (ii).
If the data are not multivariate normal and i.i.d. then the variance-covariance approach will not reflect true risk. For example, if there is serial correlation then risk is understated.
Your intuition around the use of the central limit theorem can be applied by using a bootstrapping approach to estimating VAR. This approach treats VAR itself as a random variable which is estimated with confidence. VAR has some highly unusual properties since it is not a coherent risk measure - so good luck! There are various posts on this site on how to do bootstrapping.
If you are not taking a mean of many values (with finite variance) then the central limit theorem does not apply. To calculate VaR anyway you can start taking the empirical quantile or use more sophisticated estimators, as the other answer mentioned.
Sorry but it's not clear to me the role of those different dependent variables.
PS: please distinguish between VaR (Value at Risk) and VAR (vector autoregression)