Hi Kimi,
In general if you're pushing INTO Marketo, you'll want to use the API instead of Webhooks. Webhooks go OUT of Marketo and to another system.
The real question here is how can you do something in the support system that makes an API call. If you were able to use the SOAP API, it would be a matter of making the syncLead call and passing in that lead's information. It should be pretty simple, unless adding things to the support system is intrinsically hard,