This is all very early in the long journey towards sainthood, and we're still learning the canon law process, but from what we can tell the first step in petitioning for the opening of a Cause for Beatification and Canonization is for there to exist an "opinion that has spread among the faithful about the purity and integrity of life of the Servant of God and about the virtues practiced by him to an heroic degree." By sharing quotes by Tolkien and talking about the example of his life, we hope to see this this opinion spread.
Next a group called an Actor Causae is formed, which will submit, through a postulator, a formal request for an investigation to the bishop of the diocese where John RR Tolkien died. Once the Actor Causae submits a request to the local bishop, the bishop conducts an investigation to determine whether or not the evidence is substantial enough to make a request to Rome.
The Congregation for the Causes of Saints reviews the local bishop's request to open the cause. Once the request is approved, the local bishop opens a tribunal. We will be updating this site with more information regarding the rest of the process as we learn more.