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Being new to AOS, can someone confirm...

My SCE Vanquishers have a Lightning Strikes ability, adding 2 to their attacks characteristic if fighting a unit of 10 or more. Their swords already have 2 attacks, so each man now attacks 4 times if they're fighting, say a unit of 10 nurgle plaguebearers.
A standard unit of Vanquishers is 5 strong, but if I reinforce the unit I can have 10 vanquishers.

Now...does this mean I'm rolling 40 dice to attack? If so, this is a mighty onslaught!

Yep, you're correct.

As for how effective all of those attacks are, well, it depends. It'll clear out lesser chaff, like goblins or skaven clan rats. Plaguebearers are tougher customers, however.

IIRC, Vanquishers are a popular battleline choice for SCE, because they're relatively cheap in squads of 5 and can put out a respectable punch.

Entirely correct. But before exclaiming, consider that a similar point unit of plague monks or bloodreavers is rocking 90 attacks when their respective buffs are triggered. There are many units which can throw out an inordinate number of attacks, but as Thadin wisely mentioned above, the quality of those attacks matter.

In this specific case you'll need 3 successful wounds to statistically kill a plaguebearer; those attacks are only killing ~6 of them on average. Now against plaguebearers that is actually pretty good, but it goes to show how statistics can really slim down the numbers once factored out.