Whether or not the addition of a new contact can trigger a workflow depends on which API call is being used from Leadsbridge. If the members/add call is being used then that won't activate a signup trigger.
You need to make sure that a members/signup with a signupform_id parameter call is being used. This will be treated as though that member has used your signup form and thus be able to set off a signup-based trigger, as well as a confirmation email (should you wish).
On looking around on Leadsbridge I wasn't able to find any clarifying documentation surrounding their API and which call they use, most likely if you're currently testing it and that workflow isn't triggering it's because the wrong API call is being used.