Within Xero you have Contacts and you have Suppliers. Lightyear recognizes these as two different types of records. If you try to create a supplier within Lightyear and it does not work, the likely cause is the contact already exists in Xero.
When you create a Contact, it does not automatically become a Supplier until you assign a purchase bill against it. Whilst a contact is in the 'status' of contact, it will not sync back to Lightyear as a Supplier. However, it will prevent you from creating that supplier in Lightyear as the app detects a contact already exists in Xero and helps you avoid creation of duplicates.
Moving forward you should always create new contacts in Lightyear and Lightyear will then create that contact in Xero when we send the first Bill across.
If you have already created Contacts in Xero and need to get them into Lightyear, you have a few options
Raise a dummy bill against the Contact in Xero (eg: 1 cent) and then delete the bill to convert the Contact into a Supplier. Then head back into Lightyear to sync your suppliers across
Archive the contact completely in Xero. This will mean you free up the Contact name which allows you to create the supplier from within Lightyear. This will sync across to Xero upon first bill export.
If you're still fairly new to Lightyear and perhaps recently migrated to Xero, you could consider "Import all Contacts" and Lightyear will treat these as Suppliers.
Warning: If you Import All Contacts, these will bring through ALL your contacts. You may wish to review your Contact list before taking the third option.