I am a senior at the University of Vermont looking to move to Montreal next year and get a job. It looks like I will have to apply for an open work permit.. Will I need an employment offer before being eligible to work there? I have no close family members with citizenship, however, my grandma was born in Canada.
Based on the information you've provided, you don't qualify for an open work permit.
Assuming you are American, if you are able to secure a full time job offer from an employer in Canada for an occupation that falls on the NAFTA list, then you can obtain a closed work permit (tied to that specific employer) without the employer also needing to obtain an approved LMIA. Otherwise the employer needs to obtain an approved LMIA and give you the job offer in order for you to qualify for the closed work permit.
I'm assuming you must have been looking at the post graduate open work permits. If so, those are only available to those who have studied in Canada and therefore not available to you.