Yes, you can try applying for a work permit.
You will first need to find an employer who is willing to hire you. That employer will then need to obtain an approved Labour Market Opinion. As part of this process, they will have to prove that the job was advertised and that they could not find a Canadian to hire for the role. Assume this step of the process will take several months to complete. There is no guarantee the Labour Market Opinion will be approved. Many/most employers are not willing to go through this process because they have to pay a fee, because it takes a long time to process the paperwork and because there is no guarantee of approval.
If you are able to find an employer who is willing to hire you and that employer is able to obtain an approved LMO, you can then try application for a work permit. As part of this process, you will have to prove that you are coming to Canada as a genuine temporary worker and have no plans to overstay your visa in Canada. Note that not all work permits are approved. And the challenge you might face is that CIC could suspect you are just trying to find ways to say in Canada long term because you first came here on a student visa (and didn't complete your studies) and are now trying to remain in Canada on a work permit.
If you want to go the work permit route, I would recommend that you read through the Foreign Workers section of this forum and make sure you take a look at some of the threads where works permits have been refused.
In the end, yes - switching from a student visa to a work permit is possible. However none of us here can tell you if you will be approved. That will be up to CIC.