NKUNTESE, Ghana—Beside a coconut farm in this lush mountain town squats the signature endeavor of Ghana's Space Science and Technology Center. It is a satellite dish, rusted and infested with bees, that technicians hope to convert into a stargazing radio telescope.

In May, this West African country became the latest on the continent to set up a space program, highlighting how governments here are turning to private companies and regional economic power South Africa for a lift...