Incident ticket opening

Answer:
It's important that all incidents are recorded, no matter where they come from (the source of an incident). Source could be internal support (any level) or external (as you mentioned - e.g. supplier). For example, if supplier is responsible for certain functionality in scope of the IT service you provide, their task will be to maintain it. That also includes recording incidents if they see some e.g. malfunctions.
See the articles for further details:
"Incident Management in ITIL – solid foundations of operational processes" https://advisera.com/20000academy/blog/2013/05/21/incident-management-itil-solid-foundations-operational-processes/
"Incident Record – you can’t live without it" https://advisera.com/20000academy/blog/2014/07/01/incident-record-cant-live-without/
or visit free webinar
"ITIL Incident Management Process Demystified" https://advisera.com/20000academy/webinar/itil-incident-management-process-demystified-free-webinar-on-demand/