Should be able to do it when creating an Apple ID while setting up a new device. Step 8 in this guide: https://support.apple.com/en-us/HT202033

It does not seem possible to create an iCloud-only Apple ID after the fact.

However, you can create such an Apple ID during the initial setup process of an iOS device or macOS user account.

Create an Apple ID when you set up your device

4. Tap Use your current email address, or tap Get a free iCloud email address.

https://support.apple.com/en-us/HT204316

I've confirmed that this works from both iOS and macOS. Wipe an iOS device or create a new macOS user account and you will be allowed to create a new @icloud.com address rather than provide a 3rd party email address.

Each Apple device additionally seems to have a strict limit for the total number of iCloud Apple IDs it can create: Why won't my secondhand iPad let me activate an Apple ID?

Now I see I was mistaken about Apple requiring an existing email address. It is possible to set up an Apple ID without an existing email. The option exists when installing OS X for the first time. There I see the option to create a new Apple ID. When creating the new Apple ID the option exists to get a new @icloud.com email address. This makes much more sense. I would assume the option also exists from the iCloud System preference pane.However, I will still have to live with my two Apple IDs, one being my name and the other being my name followed by an @ sign.

How are you here posting on Reddit but don’t have (or have the ability to create) an email address?

Posting this here in case there's anyone else out there who doesn't use Apple devices, but needs to create an Apple ID and access the Apple app store.

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I eventually succeeded in creating an Apple ID via the Create Your Apple ID web interface. (It seems Apple didn't like something about the phone number I was initially using to confirm my account… but not sure what.)

After that, however, I faced a new problem.

I want to use ipatool to extract iOS software package distributions from the Apple app store, but after authenticating, ipatool was giving me errors like the following…

$ ipatool search "keywords on app store"
ERR error="you have not used this Apple ID on the app store before"

It seems Apple really wants to prevent me from accessing the app store until it thinks I have an actual Apple device linked to my account. (Eesh, what control-freakery .)

So then I tried logging in to my new Apple ID on the web, and browsing the web interface for iOS apps on https://apps.apple.com … but this still didn't enable ipatool to work.

However, I eventually discovered a solution.

I visited https://music.apple.com, and used my new Apple ID to sign in, and… then ipatool started working with my Apple ID.

Happy hacking.