Location of the Vast Gate
Far Realm was initially created by Bruce Cordell for the AD&D 2e module The Gates of Firestorm Peak. This module is dated 1996, so you cannot find any lore from, say, Planescape series of products. In the module, we learn that some long time ago, a so called "Vast Gate" was opened by Elder Elves and then had to be sealed, yet the sealing was imperfect and was re-opening periodically under certain conditions and the players try to close it completely.
The setting is described to be located near the Firestorm Peak, in "the Shirelands, at the southern foothills of the Mountains of Frost". Such a location is not defined in any of the official campaign settings of D&D. Cordell penned a number of different products at the time and these products occasionally refer to the same locations or people (forming something like a Cordell-verse), but each of these products were meant to be easily integrable to any campaign setting, so we really don't know when and where the Vast Gate was first opened within the D&D multiverse.
On the other hand, a number of people have compared some of Cordell's later work, which takes place in Greyhawk (3e's default setting), with the Cordell-verse generic products and concluded that there are both for and against arguments for integrating the Vast Gate into the history of Oerth. An example proposal that discusses the available data can be found on a 2010 post by user ripvanwormer on planewalker.com. Taking into account the date when the elves first came to Toril, he proposes that the Vast Gate could have played a role in that migration and hence considers sometime in the middle of -30,000 to -18,000 DR to be reasonable time for the accidental first contact of the Elder Elves with the Far Realm.
After the Closing of the Vast Gate
The Vast Gate is revisited in Cordell's article "Enter the Far Realm" published in Dragon magazine #330 (2005). At this point it is described to be no longer open. Yet occasional contacts with the Far Realm are said to happen at random locations, causing so-called "cerebratic blots" to appear in the Prime Material Plane and allowing travel. Yet again, no mention of any of the official D&D campaign settings is made, except for Eberron.
Likewise, if we look at the discussion in the 3e Manual of Planes:
There are no known portals to the Far Realm, or at least none that are still viable. Ancient elves once pierced the boundary of eons with a vast portal to the Far Realm, but their civilization imploded in bloody terror and the portal's location is long-forgotten. But other portals might exist.
Other methods of reaching the Far Realm include traveling to time's beginning or end or finding the true Dreamheart past the Portal of Sleep.
The 4e Manual of Planes does not contradict with this description, mentioning that it only touches rarely the material world and that it is not normally a place one can visit.
Novels - Living Gate
This lore is somewhat contradicted by the Forgotten Realms short story Watchers at the Living Gate by Paul Park published in the D&D anthology Untold Adventures. It describes the so-called Living Gate, a tiny portal located in the Feydark, beneath the ruined eladrin metropolis of Cendriane. Cendriane was mentioned in the 4e Manual of Planes, but to the extent that I could track, the Living Gate was not mentioned anywhere beyond the short story.
Eberron Specific
Finally, if you are playing in the Eberron campaign setting, the Far Realm is incorporated as the Realm of Xoriat in Cordell's above mentioned Dragon article. Yet, please note that Eberron exists in its own micro-cosmology, so it is up to the DM how to interpret the Far Realm - Xoriat correspondence. [There has been a motto that everything that is in D&D is in Eberron but not possibly in a form you recognize.]
