I've recently been reading 'Nathaniels Nutmeg' and from it it appears that, among others, James Lancaster used lemon juice as a cure for scurvy in 1601. This was reported to the Admiralty, but nothing came of it, and the curative power of lemon juice was not widely known until 1747 and not adopted by the British navy until 1795. It is estimated that perhaps a million people died of scurvy between these times.
Look at
http://www.riparia.org/Medical%20History/scurvy_hx.html said:In 1601 Captain James Lancaster unintentionally performed a controlled study of lemon juice as a preventive for scurvy. His fleet of four ships departed an April 21, 1601, and scurvy bagan to appear in three of the ships by August 1 (4 months after sailing). By the time of arrival, September 9, the three ships were so devastated by scurvy that the men of Lancaster's ship had to assist the rest of the fleet into the harbor. Lancaster's men remained in better health than the men on the other ships because he brought to sea bottles of lemon juice, which he gave to each one as long as it would last, three spoonfuls every morning. The Admiralty received Lancaster's report.
Note that even in his ship, he didn't ELIMINATE the scurvy, but just reduced it. An interesting datum, but not proof.
OTOH, I like your idea of the EIC instigating the use of citrus. They presumably have a reasonable source of supply in the Indies (that's where citrus is native to). They probably start with the fruit, as long as it lasts, and then try various means of storing the 'anti-scorbutic'. Glass bottles, which is what Lancaster seems to have had, may be good if they can be sealed properly, but they'd be horrible to try to keep in a storm. You'd need gallons or hogsheads of the stuff for any decent sized ship for any length of voyage.
You might want to check prices, and see how economical it would be...