In June, Amazon announced a new business it says almost anyone with enough savings and a desire to be their own boss can do: staffing and operating a fleet of delivery trucks for the e-commerce giant.

It's a program called Delivery Service Partners, and Amazon is hoping it will solve the problem of last-mile delivery. With an initial investment "as low as $10,000," according to Amazon, an entrepreneur can start a business that hires drivers and leases up to 40 vans to deliver packages from warehouses to homes.

Since it was announced June 28, tens of thousands of people have applied to the program and Amazon has put in an order for 20,000 Mercedes-Benz vans from German automaker Daimler, according to The Wall Street Journal.

"We have been humbled by the tens of thousands of individuals who have applied for the program," an Amazon spokesperson tells CNBC Make It.

For the program, drivers will wear Amazon-branded uniforms and the vans will sport Amazon Prime logos — but the business will be owned by the entrepreneurs themselves and contracted by Amazon to complete deliveries, according to Amazon. (The company won't allow vans with Amazon branding to be used to service competitors, according to The New York Times.)

To keep the start-up costs low for interested owners, Amazon is providing access to third-party deals on costs like leasing the vans, getting insurance and buying mobile devices with data plans. Although the newly formed businesses aren't required to use the deals through Amazon, they "may not be able to achieve the start-up cost figure [estimated as low as $10,000] without doing so," Amazon discloses in a brochure about the program.

For "successful owners" operating with 20 to 40 vans, Amazon estimates potential annual profits will range from $75,000 to $300,000, although those numbers will differ by city and the individual business' costs, according to the brochure.