Answer: You are correct that schools cannot ask parents for documentation of their homeless status as a condition of enrollment. The relevant citation is 42 U.S.C. § 11432(g)(3)(C), which pertains to immediate enrollment:
(C) IMMEDIATE ENROLLMENT-
(i) IN GENERAL.– The school selected in accordance with this paragraph shall immediately enroll the homeless child or youth, even if the child or youth
(I) is unable to produce records normally required for enrollment, such as previous academic records, records of immunization and other required health records, proof of residency, or other documentation; or
(II) has missed application or enrollment deadlines during any period of homelessness.
In addition, state and local educational agencies are required to review and revise policies to remove barriers to the identification of homeless children and youths, and the enrollment and retention of homeless children and youths. That citation is 42 U.S.C. §11432(g)(1)(I). A policy requiring homeless families to produce hotel receipts or eviction notices creates barriers to enrollment (and possibly also identification and retention).
NCHE has produced a brief on confirming eligibility that may be helpful. It may be downloaded here: https://nche.ed.gov/wp-content/uploads/2018/10/conf-elig.pdf
One circumstance in which substantiation of homeless status might be necessary is in the course of a dispute over eligibility. If a school has reason to suspect that a family is fraudulently claiming homelessness, the school must provide a written explanation of its reasons, and include information about the right to appeal. If the family chose to appeal, they might provide receipts or notices as part of that process.