I was just looking into this problem myself, and as far as I know there are two possible ways:
Create a Facebook Page not an Account It sounds like you are using a Facebook "Account" (a person) for your Website. First of all, that is a violation of their terms of service because a Facebook account must be for a "person". Second, Facebook does not allow ANY means of login other than through their servers, so a person, using a browser, must log in through facebook.com (which, for third party applications, usually happens in a popup) and then that authorization token will be sent to the third-party application for use with APIs.
A Facebook page is perfect for this, because a Page doesn't need to be for a person only, and all content may be public. If the content is public, then you don't need an access_token at all.
Request "offline_access" permission An application may request offline_access permission for a user. This means the token will not typically expire for a long time. You will have to log in once, but then that access_token can be used for a long period of time. Still, it sounds like you are using a Facebook profile/account which is not for a person and you shouldn't be doing that anyway.