Yes, RRH programs may request bank statements and inquire about spending habits.
However, RRH providers may not REQUIRE bank statements nor make handing over bank statements a condition for housing only if a program participant agrees to this review.
It can be offered as a service to assist the client with their spending habits and better budget for rental payments, which should be emphasized since RRH is not a forever intervention. If a client does not wish to share their bank information or spending habits with the program provider, they may not threaten to remove them from the program. This could be framed as a service to help a client with budgeting, not a compliance requirement.
Peering into someone's spending habits is highly intimate, and clients reasonably expect these matters to remain private. RRH providers are not required to protect clients from eviction by private landlords, there can be natural consequences, the program just may not impose artificial "program only" consequences.