Since the Cunninghams are poor in To Kill a Mockingbird, how do they pay Atticus, who is a lawyer, and the doctor, for the services they have...

The Cunninghams pay Atticus in farm goods.

The Cunninghams are a poor but respectable family in Maycomb.  They have a lot of pride, and a Cunningham never borrows money that he can’t repay.  Scout has a classmate, Walter Cunningham, who refused to borrow a quarter from the teacher for lunch.  Scout asked him home for dinner, however, and he accepted. 

When the Cunninghams need Atticus’s help with an entailment on their farm, Scout asks Atticus if the Cunninghams will ever pay him for his law services.  Atticus tells her that the Cunninghams can’t pay cash, but they will pay eventually.

One morning Jem and I found a load of stovewood in the back yard.

Later, a sack of hickory nuts appeared on the back steps. With Christmas came a crate of smilax and holly. That spring when we found a crokersack full of turnip greens, Atticus said Mr. Cunningham had more than paid him. (Ch. 2)

Atticus has to accept payment for services from farmers in the form of goods like wood and chickens.  This is because he knows that they can’t pay.  It is the Great Depression, and everyone is poor.  It is an especially bad time for farmers.

Atticus said professional people were poor because the farmers were poor. As Maycomb County was farm country, nickels and dimes were hard to come by for doctors and dentists and lawyers. (Ch. 2)

Bartering for services is actually very common even today.  It is a good way for someone to get something he needs without actually paying money.  Sometimes one person has something another person needs, and that person has something he needs too.  It is a win-win situation.

The Cunninghams are generally more respectable than the Ewells, who are poor but have no pride.  Atticus respects the Cunninghams, but says the Ewells live like pigs.  Although there is the incident with the mob, the Cunninghams seem to be a good group of people.