These days a petitioner can be a collector of signatures, but, apart from this sense, the word is probably most often used these days in the common law legal sense, where a petitioner is a party filing a petition, i.e. seeking a court order.
The non-lawyer might perhaps be most familiar with the term from divorce proceedings, where the party filing for divorce is the petitioner (or applicant, depending on the particular jurisdiction's choice of nomenclature). More generally, any application for a court order or injunction (for example, a restraining order) is a petition and, hence, filed by a petitioner, as opposed to a claim in tort, for instance, which is filed by a complainant or plaintiff.
Putting all this aside, at its root, petitioner has a much less specific, less technical meaning.
The relevant definition of the noun petition at Oxford Living Dictionaries follows:
An appeal or request, especially a solemn or humble one to a deity
or a superior.
‘Nor, for that matter, COULD they regulate anyone's private petitions
to their own deity!’
‘Verse eight declares the defeat of the foul foe and verse nine is a
concluding petition to the God of Israel.’
‘The shaman is about to perform a cha-chac ceremony: a petition to the
god, Chac, to send rain.’
entreaty, supplication, plea, prayer, appeal, request, application,
So, in its most general sense, a petitioner is anyone making a humble request of any authority.
These days, due to the rule of law and the less public role of religion in Western societies, these entreaties are most commonly made to courts, but this isn't necessarily the case, and in other times and in other societies, a petition could be addressed to a local magnate or a king, to a feudal lord or a bandit chief (that is, if there is any practical difference between these). A petition can be addressed to anyone wielding power, up to and including deities.
Some examples of petitioner in use in this non-law-court, non-signature-collecting general pleading-meaning-asking sense follow:
LAST week Schumpeter met two tech tycoons who control businesses in total worth $600bn. In both cases the mayhem around them was what you would expect if Beyoncé hit town, minus the musical talent and looks. Hotel floors were locked down by the official secret service; the corridors were crammed with lines of petitioners and in one case a Wall Street boss gatecrashed the room in order to hug his idol.
"Who's afraid of disruption?", The Economist, September 30th 2017
These figures suggest that the social cost of rent seeking is staggering. But is it conceivable that a third (or more) of a country's population is engaged in rent-seeking? True, in India and other developing countries there are endless lines of petitioners at every office, but virtually all villagers, who make up 80 percent of the population, and most urban dwellers perform their usual productive tasks.
Stanislaw Wellisz, Ronald Findlay, The State and the Invisible Hand, The International Bank for Reconstruction and Development/The World Bank (1988)
Prince Zhuang (Yinlu, 莊親王胤祿, 1695-1767), the ‘Sixteenth Regulo’, had finally been able to place the corrected memorial into the hands of the emperor, and the somewhat unorthodox procedure had the desired effect: while the petitioners were waiting outside the palace’s gates, the emperor issued an immediate response stating the imperial political position on Christianity. It was in the form of a vermilion rescript written on the request itself and it was shown to the waiting petitioners...
Eugenio Menegon , "An Emperor Confronts Christianity and the Heterodox, Part II", China Heritage (2018)
Similarly pray, in its basic sense, means request, particularly a respectful request of an authority. And pray is still sometimes used in this non-religious sense in legal filings, where the prayer would be the action or judgement requested of the court that comes at the conclusion of a petition or other pleadings (another similar word!)
(This is in no way to detract from supplicant, which is an excellent answer, merely to offer an alternative with a different nuance.)