The machine was among other components and computers the woman wanted to dispose of after her husband died.
Only about 200 of the first-generation Apple computers were made.
The company recognised the value of the old computer and sold it to a private collector for about $200,000 (Â£131,000).
The recycling centre, called Clean Bay Area, has launched a web and media campaign to track down the woman so she can receive her half of the cash they got for the machine.
It has produced a short video it is asking people to share to see if they can reach the donor.
In a blogpost, Clean Bay Area said the mystery woman had dropped off the boxes of electronics in late April.
It said she had been between 60 and 70 years old, had driven an sports utility vehicle (SUV) and had left the boxes at its Milpitas recycling centre.
The centre is providing no more details of what the woman looked like, but said recycling centre staff would recognise her if they saw her.
"I remember her," said Victor Gichun, a vice-president at the Clean Bay company, who dealt with the woman when she dropped off her boxes.
Mr Gichun said he had asked the woman if she had wanted a receipt but she had said she did not.
As the centre had a backlog of donated items to sort through, the boxes lay undisturbed for a couple of weeks.
When they were sorted, the Milpitas centre's manager spotted the computer under old cables and keyboards and realised what they had been given.
Designed and hand-built by Steve Wozniak, the Apple I was the computer company's first product.
It went on sale in July 1976 for $666.66.
The machine had about 4kB of memory.
By way of comparison, the image at the top of this page is more than 42kB in size.
So far, only about 63 of those early machines are believed to have survived to the present day.
To get the cash, the woman just needs to return to the Milpitas donation centre.
"To prove who she is," Mr Gichun told the San Jose Mercury News, "I just need to look at her."
A San Francisco recycling centre is seeking a woman who may have handed over a rare Apple I computer by mistake.