Company I really want to work for has applications up, how weird would to message someone who works there on LinkedIn?

Not only is this not weird, but it's HIGHLY RECOMMENDED. Even if it's small, this is a really effective way to get in the pipeline. Don't be too hovering or invasive, but something like this is very lightweight and non-invasive: 'Hi! I just applied for a position at X and am extremely excited about the company. I just wanted to reach out and connect because I saw you worked there. If you have any advice for the process, let me know! This is really my dream company!' if you want to more aggressive you can do something like: 'Hi! I just applied for a position at X and am extremely excited about the company. This is really my number 1 dream company. Do you have any time to meet to give me some advice on the process?'