> I have noticed there are not many small companies on my school's job board.
The few small to medium sized companies on your school's job board are actually great opportunities. Those companies don't have the resources to recruit many places so they might not even be looking at any other schools or programs. There is probably a very strong alumni connection to your school or program. They likely know exactly what they're getting with those new grads and wouldn't be recruiting there if they weren't comfortable with that.
> Would you know of a place where I could find these?
One problem with small companies is that they tend to not advertise their job openings everywhere. You can usually find different ones everywhere you look. The worse they are at advertising the job, the less competition you have and the more likely you resume will be considered. So it does pay to look beyond any one job board.
> One thing I have worried about though is they might require someone who is more experienced.
This varies company by company. Some companies founded by new grads or college drop outs exclusively hire new grads. Some big companies exclusively hire experienced engineers. And there's everything in between. If they have a job posting for entry level or new grads, go for it.