Can I file my taxes without my social security number on my W2? My employer didn't include it.

You need to request a corrected W-2 from your employer. The IRS matches their records with your return and the social security number needs to be on the W-2. You need a W-2C for corrected.

what's most important is whether or not your SSN was on the copy sent to the Social Security Adm. it's they send the forward the W-2 info to the iRS.  if it isn't  then your employer is going to have to file w-2cs for every w-2 that's missing the employee SSN.