As far as I know, LastPass does not "store the key next to the lock."[1] The browser extension encrypts/decrypts locally. If you use your password file through the web site you're still downloading your encrypted DB from them and encrypting/decrypting locally (whether with the extension, or I believe they also have a pure JS implementation).
[1] Or so they say. I've never MITMed their SSL, and their software is not open source AFAIK. This is not to say someone couldn't e.g. distribute a trojaned version of their browser extensions. If you poke around the developer(s) have at least revealed the encryption method for the your DB so you can verify how it is encrypted for yourself, which is a good sign if nothing else.
[1] Or so they say. I've never MITMed their SSL, and their software is not open source AFAIK. This is not to say someone couldn't e.g. distribute a trojaned version of their browser extensions. If you poke around the developer(s) have at least revealed the encryption method for the your DB so you can verify how it is encrypted for yourself, which is a good sign if nothing else.