Need help identifying a pair of Lounge Chairs  

No chrome, but...
I'm pretty sure that's a Milo Baughman chair. Nice thrift score!

Looks like you nailed...
Looks like you nailed it....
http://www.flickr.com/photos/kinzco/7440225508/in/photostream/

Does anyone have opinions on...
Does anyone have opinions on redoing the webbing? it is saggy and rigid, but only one piece is detached.
Should I leave it as is?
Is it difficult to re-do?
Is webbing the proper term?

It's kind of hard to re-do.
The chair will be much more comfortable with webbing that is elastic, not dried out and stiff.
If you want it to be just like the original, get Pirelli webbing. You really should use a pneumatic stapler because the staples go in so much better with one, but an electric staple gun will do, I guess.
Remove the tape and old webbing and all of the staples. A Berry staple remover is really good for this--get them out enough to grip with a pair of diagonal nippers. You can get a Berry staple remover on ebay if you don't have an upholstery supply place local to you, or order one online somewhere.
Mark where the new webbing should go if it's not already obvious from previous marks or shadows. Staple one end. Stretch it and staple the other end. Cut. Always work off the roll, never cut a length and try to staple it as you won't have enough to hold onto when pulling it taut.
Do all the front to back pieces, then the side to side and remember to weave each side to side piece BEFORE YOU STAPLE THE FIRST END OF IT.
Cover the stapled ends with black tape. I think gaffer's tape is about the closest thing to the original but I'm not positive. There is a black masking tape sold in art supply stores that is ok.
Pirelli webbing isn't cheap. It might be worth getting a quote from a good upholster for new webbing. Upholsterers do this kind of thing on a regular basis and have the tools and the pneumatic stapler and all that...I'd just get a quote, at the very least.

Pirelli Webbing
Pirelli Webbing prices were through the roof when I needed some a few months ago. It used to be you could pick up a 100' roll for ~$125, but it had gone up to $3+ per foot, and all of the local upholstery suppliers / shops I spoke to said they had switched to different types of webbing (rubber or elasbelt) due to cost and availability of Pirelli webbing.
In the interest of staying true to the original, I opted to buy legit Pirelli webbing, and the best price I found was from a place in the UK (I'm in the US) called Skiddaw Upholstery. Even with shipping, it was cheaper than domestic sources. They were pleasant to deal with and have both 2" (fairly common) and 1.5" (not so common) widths.

Tedco Industries has it for
$135/100 feet right now but not sure how long that will last.
I'm also not sure if they do shipping. I'm pretty sure they will not sell yardage, only whole rolls.
http://tedcoindustries.com/

Is that pattern of the...
Is that pattern of the webbing original?
I know a lot of other chairs have gaps in the webbing...

*Spanky* FTW
That's a killer price on Pirelli webbing. Thank You kindly!