Yes, in theory it's pretty easy to do (although, of course, as RCB's answer notes, it is illegal).
According to California DMV, you can register to vote in Presidential elections in CA by mail, by following the process as noted (relevant parts bolded):
Register by Mail
If you prefer to register to vote by mail, you can:
Print the National Mail Voter Registration Form online...
When you complete your application, mail it to the address provided on the application. You will need to provide your California driver's license or identification card number or the last 4 digits of your social security number. If you do not have any of these numbers leave the field blank and the election officials will assign you a voter identification number.
The important part is that California's driver's license is OPTIONAL, and you can simply provide last 4 digits of SSN (which of course an illegal can make up) or even leave empty. You are NOT required to submit an ID by California.
Of course, there are requirements that come with the Federal form itself, so let's check them out. There are 2 relevant pieces:
Box 6 — ID Number Federal law requires that states collect from each registrant an identification number. You must refer to your state's specific instructions for item 6 regarding information on what number is acceptable for your state. If you have neither a drivers license nor a social security number, please indicate this on the form and a number will be assigned to you by your state.
As noted above, in California you don't need an ID # and the state will assign one to you if you don't provide one.
First Time Voters Who Register by Mail
If you are registering to vote for the first time in your jurisdiction and are mailing this registration application, Federal law requires you to show proof of identification the first time you vote. Proof of identification includes:
• A current and valid photo identification or
• A current utility bill, bank statement, government check, paycheck or government document that shows your name and address.
Note that nowhere here you are required to provide proof of citizenship, and allowable proof of identification is available to most people who aren't citizens (bank statement or utility bill don't require you to be a citizen) and are easy to forge (unlike a driver's license or proof of citizenship).
P.S. To address the usual responses of "well, if it's so easy how come we don't have any proof/convictions" - that's the catch-22. It's so easy, it's basically impossible to catch anyone doing so, and therefore, absent hard-to-forge voter ID requirements, there's no legitimate way to tell whether lack of successful prosecution is due to lack of illegal voting or lack of ability to catch illegally voting people. This isn't proof that there is any (or many) illegal voting going on - but it's also NOT a proof that there isn't.