As I stand on/at the ______ of a new chapter?

Further context could be of relevance here. Do you mean a metaphorical chapter, like perhaps, a new chapter of your life?

I suggest the following idiom, which you could use depending on how you want the sentence interpreted:

on the cusp

On the threshold or verge of a development or an action

[American Heritage Dictionary]

Thus, you could say—

As I stand on the cusp of a new chapter of life...

I believe you are mixing metaphors. The common phrasing "stand on the ..." ends with precipice. But that is for a situation that could become very bad very quickly. A precipice is a sharp cliff, where one wrong move could be fatal but the right moves could leave to safety.

A new chapter of ones life implies change, transition, or new opportunities.

If however you do mean that you are at a point where things could go well or poorly, you could say you are at a precipice, but perhaps you are more standing at a crossroads

Also, at a crossroads. At a point of decision or a critical juncture, as in Because of the proposed merger, the company is standing at the crossroads. This phrase, based on the importance accorded to the intersection of two roads since ancient times, has also been used figuratively just about as long. In the 1500s Erasmus quoted from the Greek Theognis's Elegies (c. 600 b.c.): “I stand at the crossroads.”

Drop the "chapter."

I'd say precipice. Standing on the precipice of a new chapter/era/etc., is a phrase often found in writings in books, newspapers, and so on.

For example:

1. Alan Wolfe, "Are you better off now?", in the New York Times,

Yet maybe we do not, as he says we do, ''stand at the precipice of a new era.''

2. Jesse Lawrence, "David Beckham’s Final Game Highlights Landmark Season for MLS & MLS Tickets", in the Huffington Post

Still, as American soccer fans stand at the precipice of a new chapter in post-David-Beckham-Experiment MLS history, Becks’s influence is still undeniable.

3. Michelle Goldberg, "New York Lockdown", in the Guardian

Now he talks as if he's standing on the precipice of a new era. "I like what happened in Seattle. But the real vision I have is what happened in Paris in 1968," he says, referring to the student uprising and general strike that convulsed the city.

I think the standard BE version is threshold

As I stand on/at the threshold of a new chapter."

OED:

1.a. The piece of timber or stone which lies below the bottom of a door, and has to be crossed in entering a house; the sill of a doorway; hence, the entrance to a house or building.

2. transferred and figurative.

b. In reference to entrance, the beginning of a state or action, outset, opening.

1834 L. Ritchie Wanderings by Seine 8 The youth, stepping proudly upon the threshold of manhood.

1877 M. Foster Text Bk. Physiol. (1878) iii. i. 389 We are..met on the very threshold of every enquiry [etc.].

Google search "on the threshold of"