You usually don’t have to read far in any serious work on Tolkien’s fiction before you’ll see references to Marquette University and the Bodleian Library. These are the two authorized repositories of Tolkien’s papers and manuscripts.
Marquette University is located in Milwaukee, WI and was not particularly dear to J.R.R. Tolkien in any way. William Ready had the foresight to purchase Tolkien’s manuscripts before they became valuable resources. Some of the materials were not shipped, as should have happened, and they are now located at The Bodleian Library of The University of Oxford.
The fortunate circumstance of this division of Tolkien’s papers is that North American scholars have easier access to Marquette and European scholars have easier access to Oxford, and both groups are equally inconvenienced for working with the other collection. The Tolkien Estate retains the literary rights to the Tolkien manuscripts and letters, so Marquette requires that you get permission to make copies.
A select group of researchers from The Elvish Linguistic Fellowship, a Special Interest Group of The Mythopoeic Society, have been publishing some of Tolkien’s literary essays through their journal, Vinyar Tengwar (“news letters”). One therefore need not seek access to Marquette University and the Bodleian library to conduct Tolkien research.
These are, however, the key resources for most scholarly Tolkien researches. Everyone, including the academics, must also use J.R.R. Tolkien’s books, Christopher Tolkien’s books, and a few other resources such as Wayne Hammond and Christina Scull’s J.R.R. Tolkien: Artist & Illustrator and Douglas Anderson’s The Annotated Hobbit.
Your local public library or university library, if it doesn’t have the full collection of Tolkien books, should be able to acquire them on inter-library loan for you. You can also buy the books, as many people have. The basic list of Tolkien books is extensive.
If you are feeling truly ambitious, Åke Bertenstam of The Tolkien Society Forodrim has compiled one of the most authoritative and reliable online bibliographies. But you may want to look for the bibliography by Wayne Hammond and Douglas Anderson.
There are few names which will be mentioned more often than Hammond and Anderson’s in Tolkien research. Many non-academic researchers are very familiar with their books. Another name you’ll come across frequently is Humphrey Carpenter, who edited the first edition of The Letters of J.R.R. Tolkien and wrote the first authorized Tolkien biography. Only Christopher Tolkien, as compiler, editor, and co-author of The History of Middle-earth stands above these scholars in providing Tolkien researchers with a solid foundation of information and resources.