What is the first instance of a Swayamvar in Hinduism? I know of a few popular ones like Draupadi's Swayamvar, Amba's Swayamvar, and Sita's Swayamvar. But what is the first instance of a Swayamvar?
Literally SWAYAM-VAR may refer simply to a girl choosing her own husband, with or without her parent's approval, and Devi Sati's selection of Shiva against her father's wishes may be the earliest instance.
But if you are asking about the Swayamvar ritual in which a father let her daughter choose her husband, the earliest example could be that of Satyavati mentioned in the Mahabharat. As mentioned in the Sundar Kaand Shlok shared by Keshav I am assuming this to be much before Sita Swayamvar.
Vana Parva SECTION CCLXLI states that when Savitri has come of age, her father, King Ashwapati of Madra asks her to choose her own husband:
And seeing his own daughter resembling a celestial damsel arrived at puberty, and unsought by people, the king became sad. And the king said, 'Daughter, the time for bestowing thee is come! Yet none asketh thee. Do thou (therefore) thyself seek for a husband equal to thee in qualities! That person who may be desired by thee should be notified to me.
Do thou choose for thy husband as thou listest. I shall bestow thee with deliberation. Do thou, O auspicious one, listen to me as I tell thee the words which I heard recited by the twice-born ones. The father that doth not bestow his daughter cometh by disgrace. And the husband that knoweth not his wife in her season meeteth with disgrace. And the son that doth not protect his mother when her husband is dead, also suffereth disgrace.
Hearing these words of mine, do thou engage thyself in search of a husband. Do thou act in such a way that we may not be censured by the gods!'
That the practice of Swayamvar was an accepted one is attested to by the Manu Smriti that mentions in the Verse 9.90 -
त्रीणि वर्षाण्युदीक्षेत कुमार्यर्तुमती सती । ऊर्ध्वं तु कालादेतस्माद् विन्देत सदृशं पतिम् ॥ ९० ॥
Having reached puberty, the maiden may wait for three years; after that time, she shall procure a suitable husband.—(90)
Verse 9.91 states:
अदीयमाना भर्तारमधिगच्छेद् यदि स्वयम् । नैनः किं चिदवाप्नोति न च यं साऽधिगच्छति ॥ ९१ ॥
When a maiden, when not given away, herself procures a husband, she incurs no sort of sin; nor does the man whom she weds.—(91)
If you're looking for the mention of a proper ceremony where kings had come to attend and test their mettle against each other, it would perhaps be Damyanti Swayamvar though I doubt its timeline in relation to Sita ji's Swayamvar.
It may have actually happened later because as per the genealogy mentioned in the Puraans, Nala belonged to the lineage of Rama's son Kush unless he is a different person altogether.