As others have stated, the CDMA EV-DO Revision A technology that the Verizon iPhone uses shares voice and data over the same connection, which wile bringing excellent call quality, hampers use of the phone while on a call. There are several workarounds for this:
Upgrade the network/hardware to utilize EV-DO Revision B, which separates the connections while increasing the bandwidth for both. Verizon does not plan to do this. Nor does Sprint.
A more practical option that you can do today is upgrade your hardware to LTE. Since they currently use two separate antennae, data can be pushed from LTE while your call is over CDMA.
An even more practical option that doesn't require a new device is connect to a WiFi network. Data will be transmitted over WiFi while your call is over CDMA.