You should probably tell the professor, or the TA, or someone. At my college, if you're not there for the first day in certain classes, you can't take the course. You should make sure this isn't one of those cases, or, if it is, ask if they can make an exception for you.
If you e-mail the professor, you can tell him/her exactly what you told us: that you have mandatory job training that can't be done any other time and that you can't have the job without. Just be honest and polite, and show enthusiasm for the course so the professor knows you aren't just trying to blow them off.