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To apply for U.S. citizenship, one has to submit the N-400 Application for Naturalization Form (mirror). After some time, the USCIS send notices the applicant for the next steps.
Is it possible to receive email notifications when the USCIS send notices for the N400 Application for US Naturalization process?
Yes.
You certainly apply online and check your status. From https://www.uscis.gov/n-400
Filing your Form N-400 online
You must create a USCIS online account to file your Form N-400 online.
Having an online account will also allow you to:
Pay your filing fee online;
Check the status of your case;
Receive notifications and case updates;
View personalized case completion date estimates;
Respond to requests for evidence; and
Manage your contact information, including updating your address.
Bullet 3 clearly states "Receive notifications and case updates"
Since you apparently travel a lot, it's worth checking if you meet the physical presence requirements.
Specifically Part 9 can be difficult for frequent international travelers, unless you keep good records of your travels. We reconstructed this using credit card bills, the "box of boarding passes", trying to decipher passport stamps, recollection, booking e-mails, etc.
We attached a "supplemental sheet" with all the details and added a note: "The following list has been created using e-mails, travel receipts, credit card records, and immigration records to the best of our abilities." That seemed to have been acceptable, but we also weren't even close of hitting the physical presence limit.
On https://www.uscis.gov/n-400, the USCIS now states even more clearly "See all case correspondence":
I did read one case where the interview notification was missing from the history, but that's atypical:
