In order to access user's account, you will need to get them to generate a token for use with your application. You can do this by sending them the link to your application, which is generated by clicking the 'Open install URL'.
When a user opens this link, and installs your application, they will install the latest manifest. Therefore, you will need to make sure that you have saved your manifest, created a new version, then set the version you have created to live via the 'Set live' next to the manifest version.
In order to generate an access token (A token that you can use any time of the day, in any place -- Usually used in mobile applications, or applications that require access to the account when the user is not using the system), you will need to make sure that you have a module with the type 'ExternalApplication' in your manifest. You can have multiple modules, but one of them needs to be an ExternalApplication. This tells the authorization that the user requires an access token, and will generate one when they install.
You can request the user to give you this token, which you can use in conjunction with the AuthorizeByApplication call (See apps.linnworks.net documentation). This will generate a session for you to use through the Linnworks.net API.