Is the following sentence grammatically correct and natural?
I don't work yet as I am a student.
If I were a native speaker of English, I would never shut up. :-)
Plenty of people are students and also have a job so, although your sentence is grammatically OK, you can't really suggest that the reason you don't have a job is simply because you're a student.
I wouldn't give one as the reason for the other. I would simply say:
I don't work/I don't have a job. I'm a full-time student.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.