Is there a requirement for the Catholic Church to file an EEO-1 Report?
Legally, there is no “Catholic Church”
The Catholic Church in the USA is a hodge-pudge of legal structures (predominantly trusts, corporations sole, and regular corporations) governed by disparate state law. In most cases, each conceptually distinct part of the church - parishes, bishoprics, schools, seminaries, cemeteries, universities, holy orders etc. - is its own legal entity.
Whether each part needs to respond to EEO-1 reporting depends on if it individually meets the 100 employee threshold. The overwhelming majority won’t meet the threshold. Those that do, need to file a report subject to the religious category exemption.
29 CFR Part 1602 says (§1602.7)
every employer that is subject to title VII of the Civil Rights Act of 1964, as amended, and that has 100 or more employees shall file with the Commission or its delegate executed copies of Standard Form 100
which is "EE0-1". The Catholic Church is not fully exempt. The requirement does not apply
to a religious corporation, association, educational institution, or society with respect to the employment of individuals of a particular religion to perform work connected with the carrying on by such corporation, association, educational institution, or society of its activities.
There is no exemption for reporting as regards race, color, gender, pregnancy, or national origin.
Picking one state as an example (Washington), there are three corporations corresponding to the diocese of Seattle, Yakima and Spokane. It is hard to interpolate from known resources to employees, but there are almost 200 Catholic churches in the Seattle diocese, and 75 schools, so it is likely that this diocese passes the 100 employee threshold.