Oyston apologised earlier this week after a series of derogatory texts he sent to Blackpool fan Stephen Smith were published online.
Oyston called Smith "a retard" and told him to ''enjoy the rest of your special needs day out''.
Smith has sent the transcript of his heated text exchange with Oyston to the Football Association, which will decide over the coming days whether to charge the Blackpool chairman.
Smith has rejected Oyston's apology and is now demanding the Tangerines chairman resign and allow club president Valeri Belokon to buy the club.
The petition, launched on change.org, was set up on Christmas Eve morning at around 10am
Within four hours, 1,000 people had signed the petition, which called on Oyston to "step down from any position of influence in Blackpool FC immediately".
The text in the petitions says Oyston's comments make him "unsuitable to hold a position of influence within our famous community football club."
Oyston's father Owen bought the club in 1987
His son has been chairman since 2000.
Belokon owns a 20 per cent stake in the club
On Tuesday the Latvian called on the Oystons to relinquish their ownership of the club in light of the latest controversy surrounding the family's reign at Bloomfield Road.
Belokon told BBC Radio 5 Live he was "in shock" at the nature of the messages, which he said were "unacceptable".
Belokon said he would be interested in taking over the club, but only if the Oystons were removed.
Smith has given his backing to Belokon.
"I 100 per cent support Valeri," Smith told Press Association Sport.
"I met Valeri in March to talk about the future of the club as I thought it was going in the wrong direction.
"He was a real gent
He talked about how he thought the football club should be run - as a community asset, there for the people, fully transparent..
he said there would be no closed doors, and that's why I support him."
Even though Blackpool were promoted to the Barclays Premier League in 2010, Oyston has endured a rocky relationship with many of the club's supporters.
They now sit bottom of the Sky Bet Championship
Last weekend they lost 6-1 to Bournemouth.
"I think the Blackpool fans will be more elated if Oyston goes than they were when Blackpool got promoted to the Premier League," Smith added.
"That's the depth of bad feeling towards him in the town."
In October, fans staged a walkout in protest against Oyston's ownership and fans are likely to voice their dissent during the club's next home game against Rotherham on December 28.
"A few more people might turn up for the next home game," Smith said.
"It will put more money in his pocket but given what has happened, the feeling is fans actually want to go to the game to express their anger and I think that's what's going to happen."
Oyston said in his text row with Smith that he would ban the Blackpool supporter, but he intends to go to the game.
"I haven't had any confirmation off him whether I am banned or not so I think I might try my luck and see what happens," he said.
Blackpool are yet to respond to requests for a comment on the matter.
Source : PA
Source: PA