Hi Rion,
I tried this, setting a text custom field named "AssetName", and it worked:
http://myurl.com/api.aspx?key=xxxxxxxx&action=AddTicket&Subject=testAPI&Description=testAPI&Email=amiller@127.0.0.1&CustomFields={"customFields":[{"name":"AssetName","value":"Asset Name 3"}]}
I received:
{"TicketID":2045,"Subject":"testAPI","Description":"testAPI","CreationDate":"2021-09-24T10:05:37.087","Type":"Help desk","Priority":"Media","State":"Open","Team":"IT Support","Agent":"","AgentID":0,"User":"Anne Miller","UserID":26,"FollowUp":false,"AgentInitiated":false,"Personal":false,"CustomFields":[{"Name":"AssetName","Value":"Asset Name 3"}],"Files":[],"Success":true,"Message":"The creation was successful. "}