In order to use the Lightspeed integration two things need to happen.
1) you need to CONNECT your BRM to your Lightspeed.
Follow the steps here, so that your BRM knows your Lightspeed account, and vice versa.
When you are done it will look something like this:
2) EACH BRM USER also needs to connect so it is granted permission to access your Lightspeed store from within BRM (and hence use the integration)
When you first setup the connection, in step 1) above, that user is automatically connected.
But for each additional BRM user wishing to use the integration, please visit the settings -> integrations -> lightspeed page and press the 'Connect Lightspeed' button.
This diagram depicts that your logins to BRM and Lightspeed are independent, but we keep track of each BRM user that has connected to LS:
See also