We're not able to set custom fields through the POST/PATCH APIs:
this seems related to the rest-openapi.yaml where property "customFields" of model "Cart" is declared as "readonly".
Steps to reproduce:
create a new custom field (input field) named "testField", for Commerce Order entity
create a new site based on Speedwell Accelerator
create a new user with the minimum required fields (no roles/organizations/user groups assigned to it)
create a new Commerce Account (Personal) and associate it to the user created before
with Postman (or any other http client) invoke this API authenticating it with the user created before
go to Order Management - Orders section of control panel and check the tab "Custom fields" of the newly created order. The custom field "testField" is empty and not filled with the value "test" passed in the API call.