Help With Vitra Eames Lounge Chair  

It's called a "shock mount"
There are four of them on the chair and they dry out and crack over time, often resulting in damage to the lower back panel.
Yours all need to be replaced by someone with some experience in these types of repairs.
Try a search on this forum or elsewhere on the web for "Eames lounge shock mounts" and you'll find plenty to read.

You can send them to Vitra...
You can send them to Vitra to repair them.
Nearly all homemade glue restorations will break after a time.
It is not cheap, but if the wood isn´t broken it can be done.
I also don´t think your one is from 1950s.
Is there a Vitra sticker on it?
I would say mid 70s or later.

Thank You for Your Feedback
Thank you tktoo and maggy for your help. This set does not have Vitra tag or logo attached to it. I bought it from a couple in California who inherited it from the wife's father who was serving in army base in Germany. They told me that he bought it in Germany early 1960's and brought it over in mid 60's to US. That is why I assumed this was from 1950's. Any information and education regarding Vitra from that period is very much appreciated especially interchangeable repair parts Vitra vs US manufactured Herman Miller - are they the same?

I'd almost bet that the shock mounts are interchangeable.
Almost... because I don't know for sure. Screw threads could be metric as opposed to English as on Herman Miller editions and spacing may differ.
It's probably best to ask an expert like Alfie at Hume Modern in LA or another that specializes in these repairs.
One thing I would recommend that you do immediately is to disassemble the other arm assembly and remove the back from the chair. It's not stable with only one side attached and is at risk of more serious damage.

Hi We r 1
I have a similar chair, with similar problem. Were you able to find out the brand and age.
Thanks!!