Yes, in theory it's pretty easy to do (although, of course, as RCB's answer notes, it is illegal).
Step 1: Registering to vote (by mail)
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. This is confirmed by California-specific instructions on the form itself (page 4)
- ID Number. When you register to vote, you must provide your California driver’s license or California identification card number, if you have one. If you do not have a driver’s license or ID card, you must provide the last four digits of your Social Security Number (SSN). If you do not include this information, you will be required to provide identification when you vote.
Now, then there are specific instructions for registering to vote by mail:
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
• 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).
Step 2: Voting
The second line of defense against voting illegally is usually the state level requirement that those who registered to vote by mail and never voted before show some ID when going to vote the first time. Let's see how California handles this, as noted in California's Secretary of State web page:
What to Bring to Your Polling Place
In most cases, a California voter is not required to show identification to a polling place worker before casting a ballot.
However, if you are voting for the first time after registering to vote by mail and did not provide your driver license number, California identification number or the last four digits of your social security number on your registration form, you may be asked to show a form of identification when you go to the polls. In this case, be sure to bring identification with you to your polling place or include a copy of it with your vote-by-mail ballot. A copy of a recent utility bill, the sample ballot booklet you received from your county elections office or another document sent to you by a government agency are examples of acceptable forms of identification. Other acceptable forms of identification include your passport, driver license, official state identification card, or student identification card showing your name and photograph.
You can review the complete list of the acceptable forms of voter identification to use at a polling place when voting for the first time or call the Secretary of State's toll-free Voter Hotline at (800) 345- VOTE (8683) for more information.
As you can see, you are not required to prove your citizenship here either. You can bring utility bills. You can bring student ID cards (which, obviously, don't require one to be a citizen).
Full list of allowable documents (both photo IDs as well as address proofs) are listed in "§ 20107. Standards for Proof of Residency or Identity When Proof Is Required by Help America Vote Act" linked above;
(d) For purposes of this regulation, proof of residency or identity, shall consist of presenting an original or copy of any of the documents described below in either paragraph (1) or (2).
Note the "either" - meaning it's enough to present a proof from #2. So what's included? Things like:
(1) Current and valid photo identification provided by a third party in the ordinary course of business that includes the name and photograph of the individual presenting it. Examples of photo identification include, but are not limited to, the following documents
- (E) credit or debit card; (G) student identification card; (H) health club identification card; (I) insurance plan identification card
While these all are photo IDs, none of these are restricted to citizens. But, as per above, you don't even need a photo ID.
(2) Any of the following documents, provided that the document includes the name and address of the individual presenting it, and is dated since the date of the last general election...
- (A) utility bill; (B) bank statement; (C) government check; (O) identification documents issued by governmental disability agencies;
(P) identification documents issued by government homeless shelters and other government temporary or transitional facilities.
None of these are restricted to citizens. Some are easy to forge (especially A and B).
Step 2-Alternate: Vote by mail.
As a bonus, you can avoid the pains of showing up at the polls, and vote by mail. Let's see if the requirements for that are any more stringent, courtesy of California Secretary of State web page:
Voting for the First Time
When you registered to vote, you were asked to fill in your driver license number, California identification number, or the last four digits of your Social Security number. If you did not include this information when you registered, send a photocopy of some personal identification with your Vote-by-Mail Application or to your county elections official before the election.
A copy of a recent utility bill, the county Voter Information Guide you received from your county elections office....
For more information on identification to use when you vote for the first time, check the complete list of acceptable forms of identification.
And the above paragraph links to the same exact list of voter IDs allowed as in Step 2 above.
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.