There are no official mechanics given for this in 5e; only the results of the process are briefly mentioned.
The process of elemental binding is mentioned a number of times in the Wayfinder's Guide to Eberron... but no rules are described for how this is done.
Lightning rails and elemental airships are briefly mentioned in Chapter 1, under the heading "A Magical World" (p. 6):
Here are a few examples of how magic is integrated into everyday life in Khorvaire.
The lightning rail uses bound elementals to drive a train of carriages along a path of conductor stones. The rail links most major cities and is a simple way to travel long distances.
An elemental airship uses a fire or air elemental, bound into a ring that holds the ship aloft and provides motive force. The airship is a recent innovation that is transforming the business of transportation.
There's barely more detail about them (and elemental galleons) in the "Transportation" section of Chapter 2 (p. 46), but no mechanics are provided on how they are created.
Elemental binding is momentarily mentioned again in the description of the nation of Zilargo, earlier in the same chapter (p. 38, italics mine):
Noted for Gnomes, alchemy, education, elemental binding, entertainment, precious stones
In Chapter 3, the same nation comes up again in relation to elemental binding; the "Gnomes of Zilargo" section has a "Zil Schemes" table, and the first row provides a relevant character hook:
Your family is working on a new form of elemental binding, but they need Khyber dragonshards — lots of them.
Khyber dragonshards are described in more detail in Chapter 5 (p. 113):
Khyber dragonshards have a affinity for binding magics. Elemental binding — which is behind airships, the lightning rail, and elemental galleons — requires a Khyber shard to hold the elemental. Khyber shards are used for phylacteries, planar binding, any other effects that trap or manipulate spirits. Khyber dragonshards are also used for many necromantic rituals.
A later part of the same chapter mentions a relevant magic item, the wheel of wind and water (p. 117):
When mounted at the helm of an elemental galleon or airship, this allows a character who possesses the Mark of Storm to telepathically control the elemental bound into the vessel.
Chapter 6 is about Sharn, City of Towers. It provides a number of tables for use in quickly generating a basic story set in Sharn. The table "The Hook" contains this possible plot-hook:
The PCs discover a bound-elemental explosive device in their quarters. They have three rounds to flee or attempt to disarm the device.
And finally, there's one last mention in Appendix A: Further Reading, under the "Eberron Sourcebooks" heading (p. 165, italics mine):
Magic of Eberron (3.5E): In addition to presenting new arcane and divine spells, feats, prestige classes, and magic items, this book offers new options and infusions for artificers, explores dragon totem magic and the twisted experiments of the daelkyr, sheds light on the process of elemental binding, and touches on other types of magic present in the world.
I don't own this book, but it is likely your best bet for finding more information on the actual process of elemental binding. Of course, any mechanics from it that you wish to implement in 5e may require some changes in the process of converting it to a different edition.
As the beginning of the section notes, the Magic of Eberron book is available as a PDF from DMsGuild.com (as are the other 3.5e/4e sourcebooks mentioned, presumably).
That's basically every mention of bound elementals in the Wayfinder's Guide to Eberron. There are no mechanics given for binding the elementals, only interacting with those already bound and what they might be bound to. It directs readers to a 3.5e book for more details.