What They Saw
They extracted cell components and fractionated them, separating the carriers mentioned above from nitrogenase. Then they tested the potential of NADH and NADPH to reduce these carriers, hypothesizing that these reductants are used for many other redox reactions.
This reaction seems energetically unfavorable because the redox potential of NAD(P)H is higher than that of ferredoxins, but it has been shown to happen before, especially when the NAD(P)H/NAD(P)+ ratio is high. So it's plausible.
They found that NADPH seemed to be able to reduce azotoflavin, and could support nitrogenase activity, at least in vitro. NADH did not seem to have the same activity. Adding extra azotoflavin or ferredoxin increased activity too.
There was one more factor that they didn't identify or include in their assays, one that they replaced with spinach ferredoxin-NADP+ reductase. Presumably it is some reductase in A. vinelandii that they didn't purify right. They did figure out which fraction contained it though, and it restored most activity. It could be denatured by mild heating, apparently.
Of other substrates that might support nitrogenase activity, only those linked to NADP+ seemed to be useful in vitro: malate, glucose-6-phosphate, alpha-ketoglutarate, and isocitrate. And apparently NADP+-linked isocitrate dehydrogenase is pretty common in A. vinelandii cells.
So it seems like NADPH is the donor that starts the electron transport branch to nitrogenase.