I understand Flame Blade cannot be used with Extra Attack, but if a Druid has the Flame Blade in her hand can she use it to make an Attack of Opportunity?
Rules as Written, No
Flame Blade has a specific action that uses a melee spell attack. It is not an attack action. An attack of opportunity only requires a melee attack, not the attack action, so that does not immediately disqualify it.
What disqualifies it is that the spell specifically requires an action to use (a reaction is not an action). The blade is not literally a weapon, and so it cant be used for multi-attack or opportunity attacks. It specifically requires an action to make a melee spell attack. The reasoning might be that it requires some mental ability to concentrate the blade or to hold on to, but you're not using your physicality to wield it. Whatever the reasoning, it was not designed to be used exactly as a weapon is used.
Jeremy Crawford also agrees
The intent is no.
It should be noted that you can make opportunity attacks with other weapons, provided you meet the circumstances for that to occur. Wielding the flame blade does not prevent you from making AoO, you just can't make them with the flame blade.
Obviously, if your DM allows it, there's no logical in-world reason why you couldn't take a swing with it. However, 3d6 fire damage is a pretty powerful opportunity attack.