To support all the features provided by the AstroBox on a Raspberry Pi or pcDuino board without onboard WiFi, you would need a WiFi dongle that support Concurrent Mode. This allows you to have a hotspot and connect to a WiFi network at the same time.
The image provided has support for the Realtek 8188 chip and we have tested with the following inexpensive USB dongle:
We recommend that you purchase the same for the most seamless experience. However if you are a Linux expert, you might try to configure the WiFi yourself.