Does anyone know how to do this request?


The first thing that is required is to get the access token to the API. This is described in the external documentation section.
Shortly summarized, you will have to make the following request (client_id, client_secret and Ocp-Apim-Subscription-Key placeholders must be replaced with real values):
POST https://apitest.vipps.no/accesstoken/get
client_id: <client_id>
client_secret: <client_secret>
Ocp-Apim-Subscription-Key: <Ocp-Apim-Subscription-Key>

Please note: It is correct that this is a POST request with an empty body. This is due to some technical details of the backend solutions.
The request above will return a response similar to this, with the access_token:
"token_type": "Bearer",
"expires_in": "86398",
"ext_expires_in": "0",
"expires_on": "1495271273",
"not_before": "1495184574",
"resource": "00000002-0000-0000-c000-000000000000",
"access_token": "eyJ0eXAiOiJKV1QiLCJhbGciOiJSUzI1Ni <continued>"

I am able to get a 200 response code in Postman where the headers look like this:
Click image for larger version. 

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Any help would be appreciated