Will the Verizon iPhone 5 be capable of accessing cellular data while on an active phone call?

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.

There's some more explanation of why this is at http://bits.blogs.nytimes.com/2012/09/13/iphone-5-calls-data/ (thanks to MacRumors). This was a hardware decision. Adding support would require adding a third antenna. The relevant quote

So why does Verizon’s Samsung Galaxy S III, a 4G LTE phone, juggle calls and data? Samsung added an extra antenna so that it pulls data from the 4G LTE network at the same time that it’s using another antenna to do voice, said Anand Shimpi, editor in chief of AnandTech.

Then why didn’t Apple add another antenna? It actually already has two antennas in an effort to improve reception, and it would have had to add a third antenna just for Verizon and Sprint phones to give them simultaneous data and calls, Mr. Shimpi explained. Leaving that third antenna out allows Apple to simplify its manufacturing process of the iPhone for multiple carriers. Plus, in the next two years the 4G LTE network is supposed to evolve to support voice calls, which would render another antenna unnecessary later.