Will the Verizon iPhone 5 be capable of accessing cellular data while on an active phone call? I'm considering switching to Verizon for my family's new iPhones (partly due to cost, and partly due to free hotspot capabilities).  

The current Verizon CDMA iPhone 4 and iPhone 4S can not access data while on a voice call. Has this been improved over CDMA or LTE for VZW with the iPhone 5 hardware?
Short answer, no.  Longer answer, you can access Wifi data while on a call, but not cellular data:


If you're in an area with Wifi coverage, you'll be able to talk on the phone while using data.
If you're in an area with only LTE/3G/CDMA coverage, you will not be able to talk on the phone while using data.


A statement from Verizon is over on The Verge:


  The iPhone 5 was designed to allow customers to place a voice call on the Verizon Wireless network, while letting customers access the Internet over the WiFi.


And Anand provides techy details:


  So onto the iPhone – we know definitively that the iPhone 5 definitely doesn't support either SVDO or SVLTE. It's as simple as looking at the FCC documents and the appropriate sections in the allowed and tested simultaneous transmitters section for SAR (Specific Absorption Rate) testing. There, it's spelled out that only one air interface can be active at a time, and that only one antenna can be selected for transmit at a time. There's also an explicitly called out mention to VoLTE not being supported at present. I didn't replicate the entire table of simultaneous transmission combinations which need to be tested (it is a huge table) there are no entries with CDMA Voice being active at the same time as any data mode save WiFi. This has been confirmed as well by later statements by the carriers and Apple.