The Royal Academy (R.A.) is a private art institution in Piccadilly, London. Founded with the support of George III in 1768, it hosts an annual summer exhibition open to "all artists of distinguished merit". Around 1,000 works are selected for exhibition each year. Prizes are awarded, and the works are available for sale to the public. The R.A. has been housed at various times in Pall Mall, Somerset House on the Strand, and the National Gallery; since 1868 Burlington House has been its home.
The Royal Institution (R.I.) is an independent scientific organisation, famous for its Christmas Lectures. Founded in 1799, it is located in Mayfair, London. It is less clear why paintings of the Whitby storm would end up there, except perhaps for meteorological interest.
Therefore the Whitby correspondent is probably referring to the Royal Institute of Painters in Water Colours, a society originally called the New Society of Painters in Water Colours, but renamed in 1885, in time for this reference.