Drawing on “Time Series Momentum” (Moskowitz, Ooi and Pedersen, 2012), using return series from commodities futures is a perfectly valid way to compute the correlation between the contracts. The danger in this, however, is that you may be picking up effects driven by the peculiarities of the futures markets and not the underlying commodities. To address this, the authors analyze weekly position data from the Commodity Futures Trading Commission (CFTC) to study the trading activity of "speculators and hedgers".
From the paper:
We construct a return series for each instrument as follows. Each day,
we compute the daily excess return of the most liquid futures contract
(typically the nearest or next nearest-to-delivery contract), and then
compound the daily returns to a cumulative return index from which we
can compute returns at any horizon. For the equity indexes, our return
series are almost perfectly correlated with the corresponding returns
of the underlying cash indexes in excess of the Treasury bill rate.
As a robustness test, we also use the ‘‘far’’ futures contract (the
next maturity after the most liquid one). For the commodity futures,
time series momentum profits are in fact slightly stronger for the far
contract, and, for the financial futures, time series momentum returns
hardly change if we use far futures.
Additionally, AQR publishes the aggregate commodities return data from the paper here if that's any help.