Their response is misleading w.r.t. the law, which is here. Under 24 CFR 100.202,
It shall be unlawful to discriminate in the sale or rental, or to
otherwise make unavailable or deny, a dwelling to any buyer or renter
because of a handicap of ... That buyer or renter
§100.202 says that
It shall be unlawful for any person to refuse to make reasonable
accommodations in rules, policies, practices, or services, when such
accommodations may be necessary to afford a handicapped person equal
opportunity to use and enjoy a dwelling unit, including public and
common use areas.
This is not limited to existing tenants. The HUD website on reasonable accommodations elaborates further on the reasonable accommodation requirement:
Any change in the way things are customarily done that enables a
person with disabilities to enjoy housing opportunities or to meet
program requirements is a reasonable accommodation. In other words,
reasonable accommodations eliminate barriers that prevent persons with
disabilities from fully participating in housing opportunities,
including both private housing and in federally-assisted programs or
activities. Housing providers may not require persons with
disabilities to pay extra fees or deposits or place any other special
conditions or requirements as a condition of receiving a reasonable
accommodation.
As an example, they offer "Permitting an applicant to submit a housing application via a different means", which is evidence that the interpret the law as applying to potential tenants, not just accepted tenants. The HUD/DOJ summary of the law further provides the example
An applicant with an obvious vision impairment requests that the
leasing agent provide assistance to her in filling out the rental
application form as a reasonable accommodation because of her
disability.
They also note that "A provider is entitled to obtain information that is necessary to evaluate if a requested reasonable accommodation or modification may be necessary because of a disability", and they can "request only information that is necessary to evaluate the disability and/or disability-related need for the accommodation". Then they can deny the accommodation "if there is no disability-related need for the accommodation". If denied, the prospective tenant can file a complaint, and HUD (Fair Housing and Equal Opportunity Office) will determine whether the discrimination was because of the disability.