Hello!
Thank you for your response! :)

I am not doing anything special with the request. I am just supposed to receive the request and send it away, and then receive the response and show it to the original sender.
I did look into this matter with the help of WRC. We managed to make it work with what Enrico said, using an EnsLib.REST.GenericService and an EnsLib.REST.GenericOperation.

Okay! So what finally solved it was a discussion with the WRC support. 

Within the Postman request, and the form-data; Content-type was specified as text/plain on each part of the form. That is why %request.Data("profile") wouldn't work at first. The content of each part of the form gets sent into the REST-service as a stream.
With this said, instead I had to use this code which finally worked:

set profileStream %request.GetMimeData("profile")
set vProfile profileStream.Read()

However, one thing worth noting is that when I did not specify content-type for each part of the form-data, then the original code I used worked: 
set vProfile $Get(%request.Data("profile")) .

Thanks to WRC and everyone here who tried to help! :)

Okay! So what finally solved it was a discussion with the WRC support. 

Within the Postman request, and the form-data; Content-type was specified as text/plain on each part of the form. That is why %request.Data("profile") wouldn't work at first. The content of each part of the form gets sent into the REST-service as a stream.
With this said, instead I had to use this code which finally worked:

set profileStream = %request.GetMimeData("profile")
set vProfile = profileStream.Read()

However, one thing worth noting is that when I did not specify content-type for each part of the form-data, then the original code I used worked: 
set vProfile $Get(%request.Data("profile")) .

Thanks to WRC and everyone here who tried to help! :)