Flagline replacement for Walter Lamb patio furniture  

Does it have to be local?
You can buy it from this place. They have 3/16" and 1/4" diameter, plus others.
http://www.knotandrope.com/store/pc/viewCategories.asp?idCategory=23

Thanks for the lead
I took a look at the site, but they don't have completely smooth nylon flagline, which is what the rest of my set has (everything on the site is braided). It hadn't occurred to me that this might be a DIY project, which is why I was looking locally for a repair shop. I'm afraid I wouldn't be able to get the line as tightly wound and secured as it needs to be.
However, I will do a more extensive search for smooth flagline and give it a shot!
Thanks for the lead and the DIY idea.
Loretta

Just a heads up.
Nylon absorbs water and slackens when wet. It is also susceptible to UV degradation. Polyester or something vinyl jacketed might be a better choice for prolonged outdoor exposure.

Good Advice
I live in the foggiest part of SF so it is wet most of the summer. It is possible my existing set does have poly or some kind of coating because it has been out in the fog and rain for over 20 years and is still in good shape. I just assumed it was nylon because in my research there was a write up that said they originally came in either cotton or nylon, and it's definitely not cotton.
The set has been in the family for years I think originally purchased in the 50's or 60's. I acquired it in '90, but it is possible the flagline was replaced prior to that.
I want to match what is on the existing chairs, which makes this more difficult. I had hoped to find a repair place where I could bring in a fully strapped chair and have an expert select the right flagline and install.
Will continue my on-line search. I really appreciate the advice!
Loretta