5E Realms lore does not specifically address this. 4E Realms lore, on the other hand, does. Their exact origin is subject to some debate, even among the elves.
Collectively, the elves are known as the Tel’Quessir6 ("The People"), a title that encompasses Eladrin, Wood Elves, High Elves, Sea Elves, and so on. It is 'widely accepted' that the Tel’Quessir are native to the Feywild1. It is believed that they first immigrated to Abeir-Toril over a hundred millenia ago, while the exact date is uncertain it is believed to have been prior to −24,000 DR, with some manuscripts indicating it was as early as −30,000 DR, the same time in which dragons were setting up their empires (Note: Source on this is from AD&D)4. The Wild Elves were the first to arrive.1
The exact origin of the Elvish race is subject to some debate. Some ancient manuscripts suggest that they were the result of a battle between Gruumsh and Correlon that occurred in the Feywild. During said battle, Correlon was injured and his shed blood became the Eladrin.2. Other types of elf, not liking the superiority this implies in that the Eladrin are the first and 'purest' of elves, disagree with this theory3. Another theory postulated is that since the Feywild tends to be a 'reflection' of the Prime Material Plane, and thus produces creatures that are echoes of creatures from the Prime, it is also possible that the Elves are one such echo... though what, exactly, they would be an echo of is uncertain1.
Perhaps also worthy of mentioning is the current in-lore explanation for how the Feywild (also known as Faerie) is treated according to current lore. The Feywild was created as it exists today, a 'lighter' echo of the Prime Material Plane. It was made by Primordials in the far distant past, before recorded history begins.5
For a while, the Feywild existed alongside the Prime and interaction between the two was common. But, at some point in ancient history, the two drifted apart and travel between them became increasingly difficult2. This is the explanation for why, in prior editions of D&D, the Feywild didn't "exist" in the Forgotten Realms. It was out there, you just couldn't get to it.
The Spellplague (the event that 'kicks off' 4E in the Realms) changed that, reshuffling the cosmology to drag the Feywild back into close proximity, and placing it in the accessible location it resides in today5.
Checking through published 5E Realms material, it does not contradict any of this... the 5E Realms cosmology is essentially the same as the 4E Realms cosmology, and the lore established within 4E is considered to hold true in 5E, except where directly overridden. And even then, the changes are usually justified in-lore (such as the array of gods and goddesses getting rejiggered due to Ao meddling with them).
The primary lore sources we have in 5E Realms are the Sword Coast Adventurer's Guide, the DMG, and published adventures. None of these go into detail on the origins of the Elves, or even on the specific nature of the Feywild... thus we may assume the lore of older sources hold true.
1 Brian R. James and Ed Greenwood (September, 2007). The Grand History of the Realms. (Wizards of the Coast)
2Bruce R. Cordell, Ed Greenwood, Chris Sims (August 2008). Forgotten Realms Campaign Guide. (Wizards of the Coast)
3James Wyatt (December 2007). Dragon 361: A Fractured Family. Wizards of the Coast.
4Steven E. Schend and Kevin Melka (1998). Cormanthyr: Empire of the Elves. (TSR, Inc) NOTE: AD&D source
5Dungeon Master's Guide 4th edition. (Wizards of the Coast)
6Kim Mohan ed. (2015). Sword Coast Adventurer's Guide. (Wizards of the Coast)