Why doesn't the Army offer Emergency Loans to Reservists?

AER is not the Army. AER does not receive any appropriated or non-appropriated government money. It is a private non-profit 501(c)3 charitable organization. AR930-4 covers the relationship between AER and the Army.

Who is eligible?
Soldiers on active duty and their eligible dependents.
Soldiers retired from active duty because of longevity, or retired upon reaching age 60 (Reserve Component) and their eligible Family members.
Widows(ers) and orphans of Soldiers who died while on active duty or while retired.
Medically retired Soldiers and their eligible Family members.
Members of the Reserve Component of the Army (Army National Guard and Army Reserve under Title 10 U.S.C.) on continuous active duty for more than 30 consecutive days and their eligible Family members.

Army Reserve has this program. https://www.rallypoint.com/shared-links/army-reserve-family-programs-fort-family-outreach-and-support

National Guard relief organizations vary state to state. At this link, Illinois and Mississippi had organizations right off hand. http://www.operationwearehere.com/FinancialAssistance.html

The army does not do loans for service members, although you can get an advanced pay for one month's pay that is paid back over the course of a year.
The Army emergency relief fund is not an army entity, it is a nonprofit organization, and so they can make whatever rules they choose. As everyone stated, the reason they do not offer loans to reservists is that they are not guaranteed have their funds paid back.

SSG(P) (Join to see) - Unfortunately, they financially can’t open those loans up to reservist. Because they would not have the assurance of being able to dock your pay if needed.