Tow-step authenticator without a device to authenticate

Since “all” you need to do is “browser” access into https://icloud.com/ you have a few options:

Option 1): Purchase the “cheapest” iPhone you can obtain; use it as a “code generator” (once setup, they do NOT need to be connected to perform this task)

Option 2: Depending on where you access iCloud from, you may elect to have the code sent to ANY SMS or Voice number world-wide. Obviously this requires you to know in-advance what numbers will be accessible to you.

Option 3: It might appear that Apple’s newest implementation - Security Keys for AppleID - might be a good solution.

Caveat: I’ve not used this myself … yet.

They should allow easy authentication from any browser window into your iCloud account.

Security Key Limitations:

It DOES require you to have a physical FIDO2 hardware key AND at least one backup.

It DOES require a “modern” browser supporting the WebAutn standard. (Not too much of an issue anymore)

IF you loose access to your key, you WILL need an Apple trusted device to get back in to your account (using keys deactivates legacy authentication methods)

See more here:

Two-factor authentication for Apple ID - Apple Support

Get a verification code and sign in with two-factor authentication - Apple Support

About Security Keys for Apple ID - Apple Support

It would help to know how you DO access your iCloud.

Is it w/ a Mac? If not, how?

Are you simply accessing it via the web interface using a browser via hthttps://icloud.com/

You also mentioned no iPhone. Do you carry ANY mobile phone which can receive SMS texts?

Additionally;

To clear up some other common misconceptions which you repeated in your reply.

You NEVER need a security code to use FindMy from ANY browser, anywhere.

Simply click the Find Phone button at the bottom of https://icloud.com/ in most browsers. Alternatively, using the url https://icloud.com/find does the same thing. Then ALL you need to know is your AppleID and associated password.

Lastly you stated that you DO have an iPhone. It can generate a code ANYTIME, ANYPLACE … even with no external communications whatsoever.