In general you don't need copulas to calculate VaR on portfolio. You can use historical method if you have time series of returns for the assets in your portfolio. If you have sufficiently enough data this will allow you to take into account correlation risk, non-normality of returns.
Example of code in R for equally weighted portfolio without assuming any copula or distribution (using RMetrics package and the LPP indices data provided with this packages):
lppData <- 100*LPP2005.RET[,1:6]
eqWSpec <- portfolioSpec();
nAssets <- ncol(lppData)
setWeights(eqWSpec) <- rep(1/nAssets, times = nAssets)
setAlpha(eqWSpec) <- 0.05
ewPorfolio <- feasiblePortfolio (data = lppData, spec=eqWSpec);
Target Return and Risks:
mean mu Cov Sigma CVaR VaR
0.0431 0.0431 0.3198 0.3198 0.7771 0.4472
(Note that this is most probably not the best way to calculate VaR of portfolio in R)
It worth mentioning that if you need to use copulas, you will have to do Monte Carlo VaR calculation (i.e. sample copula and calculate VaR on that data), as there are no closed form solutions available for VaR for most of the copula classes.
And yes, Gaussian copula would suffer the same problems as estimating the multivariate normal distribution. Instead of Gaussian copula you can try elliptical t-copula (but note that it's symmetric) or empirical copula. Yes, Gaussian copula and other normality assumptions are highly criticized in many papers for underestimating the tail risks.