A program may require any supporting documents necessary to confirm their eligibility for the program. They may also require any documentation that a landlord may require (which may include income verification for low-income tax credits, depending on the housing provider).
For RRH, at entry, HUD does not require that clients submit any proof of income until their eligibility is re-certified, at which point the client must meet the income eligibility requirements. However, we encourage RRH providers to ask for income statements anyway at entry to meet case management goals. If you are re-certifying the client's income for the purpose of eligibility after the client's initial term in RRH, then yes, you may require it since it would affect the client's eligibility.
As a general rule of thumb, if the document is required for eligibility, you may mandate it; if it is just useful to meet case management goals, you can ask but may not require it. If a client claims that they lost a job, whether or not they still have that job is relevant information about their income levels which, depending on the project, may be necessary to confirm their eligibility.