The Church Auditing Department tells me the Church is aware of this issue and is working on expanding the ability to pay businesses using electronic transfers. Hopefully it's high on the priority list.
As an assistant area auditor I have talked with a number of stakes about wards and branches not paying fast offerings directly to providers of goods and services. There are several things they have been able to do to stay in compliance with Church policies.
Frequently (in my experience), the reason the ward or branch doesn't pay directly is that the member has waited until the last minute to ask for help. That's a teaching issue. Reimbursement may be needed if the member has never received fast offering assistance. It should rarely happen a second time. When a member comes to the bishop with a disconnect notice, etc. it is usually because someone hasn't taught welfare principles or someone has ignored what they have been taught.
Food: Wards and branches are almost always able to set up an account with a local grocery store. They just have to talk with the store. I have not heard of any ward that isn't able to find a store that is willing to do this; this is not an uncommon situation for stores. A member of the Relief Society Presidency (or someone else the bishop or branch president authorizes) can then go to the store, buy the needed food, etc. and the store sends the unit a monthly statement. The unit reconciles the statement to identify the fast offering recipients and pays the store by check. If the unit is geographically large it may need to set up accounts with more than one store.
Hotels: Most hotels -- especially if they know the Church is paying the bill -- will set up an account. Again, you just have to ask. A conversation with the manager can also be a great opportunity to explain the Church's fast offering program. (It's also helpful, perhaps, to remind the manager that credit cards usually cost the hotel more than a check.)
Rent: Landlords, I have discovered, often don't want to accept checks because they are used to tenants bouncing checks. They say they won't accept checks but what they are really saying is, "I don't want to accept the risk of a bounced check." If they know the Church is the payor, they will usually make an "exception" to their rule.
Other fast offering payments. It is rare that other providers (utilities, medical, insurance, mortgages, phone, etc) will not accept a check. When I review audit and see a pattern of those kind of payments being reimbursed to individual members, it is my practice to remind the stake audit committee that the stake (1) may need to provide additional training to ward leaders and (2) it may want to review the monthly stake financial statement more carefully to identify wards that have multiple fast offering payments reimbursed to individuals.
Has anyone else had success with setting up accounts with providers?