I figured it out:

Instead of just calling the action directly from the form I changed the type of the button to "object" and had it call a function. Then, I created a function in hr.employee that searches for the client action by id, inserts the record's id in its params attribute and then returns it.

def action_take_picture(self, cr, uid, ids, context=None):

    if context == None:
        context = {}

    res_model, res_id = self.pool.get('ir.model.data').get_object_reference(cr, uid,
                                                                            'hr_webcam',
                                                                            'action_take_photo')
    dict_act_window = self.pool.get('ir.actions.client').read(cr, uid, res_id, [])
    if not dict_act_window.get('params', False):
        dict_act_window.update({'params': {}})
    dict_act_window['params'].update({'employee_id': len(ids) and ids[0] or False})
    return dict_act_window

I then save this value in my javascript init function:

    init: function(parent, action) {
        this._super(parent, action);
        this.employee_id = action.params.employee_id;
    },