Is it possible to operate a Nordic track A2550 treadmill without the key? If so, how?

First thing I'd try is just using any magnet. I'm sure that's all the key is normally. If that doesn't work, no you cannot use a treadmill without a key. It's a safety measure created by the company.
Here's the key for sale on amazon that goes to your treadmill.
http://www.amazon.com/Pro-Form-Treadmill-Safety-Key/dp/B002YP4KQE
Hope this helped!

Please find an overview of safety keys for treadmills here as well as a direct link to purchase some :

http://www.treadmillsrepair.co.uk/2014/08/treadmill-safety-key.html

If the above link ever stops working here is another place you can get a new safety key for the a2550