Vienna International Airport offers porter services for departing and arriving passengers. The minimum charge is EUR 15, which covers porterage for up to five pieces of luggage. More details here: https://www.viennaairport.com/en/passengers/airport/baggage_services.
The reason you cannot find an easy solution to your problem is that you are trying to do something that the system intentionally does not support.
Various sources on the Internet will claim that you are either allowed light hand luggage and one (!) piece of heavy luggage, or that you are allowed "what you can carry by yourself".
Interestingly, that ...