Northern Illinois University has a Lovecraft collection, including letters? Yup…
“The collection consists of Lovecraft’s fiction writing, letters, poems, scientific articles, pulp magazine stories, books about Lovecraft, collections of Lovecraft stories, titles of books known to have lived in Lovecraft’s personal library, and a few miscellaneous items, including several manuscript letters.”
No online finding aid to exactly what the collection holds, it seems. Fandom Directory, 2000, confirms there are only “a few letters”.
I have earlier noted down these items as being in the Northern Illinois University collection:
Letters from H. P. Lovecraft to Clark Ashton Smith [Lovecraft Collection PS 3523.O833 Z543]
• No. l, dated Aug. 27, 1932, lists “weird items in my library” (envelope included).
• No. 2, undated [1932?] discusses works in progress and subjects of mutual interest.
• No. 3, dated March 9, 1934, discusses HPL’s current reading plus comment concerning ancestry (envelope included).
Based on the call number I seem to have used this page as a source: http://libguides.niu.edu/sciencefictionsampler