in general it is just the volatility of the underlying future contract that the Asian formula transform into a average vol (based on often unrealistic assumptions about the stochastic process of the underlying future contract). I think the original Turnbull and Wakeman formula is is not valid for options on futures, but several modified version
s are so.
Naturally many challenges here. For example if there is an European plain vanilla option on the same future contract (assuming the future itself not is an average rate future contract) but this plain vanilla expire considerably time after the average option expiry. Then it is not just to take the implied from the plain vanilla and put it into the Asian option formula. In general you want to look up liquid (plain vanilla) options and make sure the asian option is priced consistent with these (but not always). Could be many things to take into account here, part art, part science ( market specific also), in some markets asian options are more liquid than the plain vanilla (? LME on some metalls?).