It was set up after 42% of A-level grades predicted by teachers were lowered by the exams watchdog.
Students wearing masks and holding placards protested against the system outside the Senedd on Sunday.
Education Minister Kirsty Williams has confirmed appeals will be allowed if "there is evidence" pupils should have received higher grades.
Plaid Cymru leader Adam Price, who addressed protesters outside the Welsh Parliament, said he was considering legal action against the exam watchdog, Qualifications Wales, and the Welsh Government over the system, which he said had "stolen" young people's futures.
In England, the exam watchdog, Ofqual is facing two judicial review cases, and Mr Price said Plaid was in talks to bring similar legal action in Wales.
"We can win and must win this fight," he said.
Jo Maugham QC, of the Good Law Project which is supporting students launching a judicial review in England, said the group was "very keen" to do the same in Wales.
Due to the coronavirus pandemic, exams were cancelled this year with students' final grades based on teachers' estimations.
Results overall improved on 2019 figures for the very top grades and at grades A* to C.
But 42% were downgraded by exams watchdog Qualifications Wales after it judged the grades were "too generous".
On Wednesday, hours before students found out their results, the education minister guaranteed that no-one would get a lower grade in their A-level than they achieved in their AS result.
Proportion of final A-level grades relating to original assessments
The petition wants all students receiving grades this year to be given those predicted by teachers, with GCSE results due on Thursday.
It said the grades process "does not treat Welsh students as individuals".
"This will disadvantage Welsh young people in their future life chances, which is unfair," it said.
Sally Holland, Children's Commissioner for Wales, has called for universities to honour offers given to students, based on personal statements, assessments, references, and interviews prior to the pandemic.
In an open letter to Universities UK, the body representing universities, Prof Holland, alongside the UK's three other children's commissioners, said "rights to an education and to fair and equal treatment have been severely compromised".
"Many students have had little chance to progress their education since those offers were made," they write.
"It is unfair to now reject individuals whose results have been arrived at by a system that is likely to have produced individual anomalies."
Plaid education spokeswoman Sian Gwenllian told BBC Radio Wales Sunday Supplement programme that GCSE students "must be really worried now, looking forward to Thursday and thinking gosh, you know, am I gonna be downgraded too?"
Some A-level students have spoken of getting results up to two grades lower than predicted, and being rejected by universities after not meeting required grades.
Ms Gwenllian, who represents Arfon in the Senedd, said: "It's quite obvious that this moderation system has thrown up all this chaos."
First year A-Level student Cai Parry, from Cardiff, who has organised the protest in Cardiff Bay on Sunday, said the grades appeals process was "too little too late" for those trying to secure university places in September.
"I hope that teacher assessed grades will be fully instated," he said.
Chris Wheatley had planned to go to the University of Southampton to study Aeronautics and Astronautics in the autumn.
He told BBC Wales he had been predicted an A* in maths the whole way through his studies, but when he opened his results he had been given a C.
"I applied to Southampton and Warwick, and they have seen that C, and they have declined both of my offers," he said.
"I've been on the phone to them for hours on end, for days now, just reassuring them, these aren't my grades, there is something happening about this."
The Welsh Parliament is on its summer break, but the Children, Young People and Education Committee has been recalled and will meet on Tuesday.
The exam board WJEC is set to outline further details on the process to submit appeals early in the week.
The Welsh Government said more than 4,000 students would benefit from the guarantee that no final grade would be lower than an AS grade.
"This is around 15% of all A-level students and makes a significant difference to the overall impact of variations between final grades and centre assessed grades," said a spokesperson.
"Even before the AS floor, 94% of the grades are the same as or within one grade of the centre assessed grades."
The spokesperson said Qualification Wales and the WJEC would share the full details but appeals could now be made where there was evidence of internal assessments judged by the school or college to be at a higher grade than the grade awarded.
Sometimes the most clever thing to say is nothing at all.