How should I approach selling a niche product or service to a large company? Do I use the "contact form" on their website? Do I call and ask to speak to a business team or salesperson?
I'm my experience both the methods you mention lead to poor results.
The best way is to be introduced personally to a person working in the large company. Look through your linkedin/facebook contacts, find somebody who is in any way connected to the company, and ask him/her to introduce you.
They will probably introduce you to a person who is not the 'economic buyer', but that is your first connection into the company. Give him/her a presentation of your service, and ask to introduce you to his/her boss.. and so forth until you reach somebody who can make the purchase decision.
Good luck! Sales in large companies take a lot of time and effort.
This was covered in question 16384 here on Startups.
To summarize you start with a live contact inside the company. You need a real live person, not email or a contact form. You use that contact to find a better contact (since your initial contact will most probably not be the right contact for the job).
You work your way through people at the company until you find someone with the actual power to recommend your product or buy it.