Short answer: yes, send the author a short note with the files. Ask for corrections to missed, misread, mispronounced or added words only, each with the chapter and the timestamp. Don't invite notes on pace or tone at this stage. Those belong in the First 15 Minutes, before the full book is recorded.
A first-time author has never reviewed an audiobook. Without guidance, corrections tend to arrive as "the bit in chapter 4 where she's angry sounded off", and you spend longer finding the spot than fixing it. Worse, an open-ended "let me know what you think" invites rewrites of the text and performance notes long after the style was agreed.
A short note sets the scope and the format in one go. Narrators who skip it often say they wish they hadn't: the book gets approved, then the change requests keep coming.
Pace, character voices and the overall reading style are agreed on the First 15 Minutes, before you record the rest of the book. Once that's approved, the performance follows it. If the author wants to change the text itself, that's a change to the manuscript, not a correction, and you can say up front how you handle those.
Chapter plus timestamp is the fastest way to find a correction. Page numbers are less reliable, because your marked-up manuscript may not paginate the same way as the author's copy. For each correction, ask for:
Subject: Your audiobook is ready to review
Hi [name],
All the chapters of [title] are uploaded and ready for you to listen to. Thanks for your patience while I finished them.
As you listen, please note anything where the recording doesn't match the manuscript: a missed or added word, a word read differently from how it's written, or a name or term pronounced differently from how you'd like. Please also flag any noise or glitch that sounds out of place.
For each one, please send me the chapter, the time in minutes and seconds, what you heard, and what the manuscript says. For example: "Chapter 3, 14:32. Heard 'the old house', manuscript says 'the empty house'."
The reading style, pace and character voices were set when you approved the opening section, so the rest of the book follows that. This review is for making sure every word matches the text.
It's easiest for both of us if you send all your notes together once you've listened through. I'll make the fixes and let you know when the corrected chapters are up.
Thanks,
[your name]
An email or spreadsheet works, but the author still has to pause, check the time, and type it out correctly. In Notetracks they leave each correction right on the waveform, so the timestamp is captured for them.

You can reply on a note to ask a question, like how a name should be said, and the answer stays on that note. When the review is done, export the notes and import them into your editor as markers, so every pickup is already on your timeline. There are guides for Adobe Audition, Audacity and REAPER.

If you're delivering through ACX, the final files still go up on ACX as usual. The Notetracks project is just where the author leaves their notes.
Chapter and timestamp are enough to find the spot in the recording. If you also want the page, ask for it as an extra, not instead of the timestamp.
Point back to the approved First 15 Minutes, which set the style for the book. Many narrators put this in a welcome document at the start of the project, along with how they handle changes to the text.
Not with Can comment or Can view. See how to let a rights holder listen without downloading.
More on proofing: how to proof an audiobook or voice-over recording.