Based on what is described in the [wiki][1] I would suggest:
`amenity=bicycle_rental`
`name=Wiener Börse`
`capacity=1`
`capacity:cargo_bike=1`
`network=Grätzlrad`
`operator=Die Wiener Börse`
`fee=no`
`website=https://www.graetzlrad.wien/bike/wiener-boerse/`
The `service` key should be used to qualify other objects (e.g. bicycle shops) that also rent bicycles. You could use that if you wanted to add the service to the operator (e.g. Wiener Börse) directly.
I also struggle a bit how to indicate that there is a cargo bike only. I would probably go for something like
`bicycle_rental=cargo_bike`
added as a subkey of `amenity=bicycle_rental`.
`amenity=bicycle_rental`. Deeper discussion on this topic should probably taken to the *Tagging* mailing list.
[1]: https://wiki.openstreetmap.org/wiki/Tag:amenity%3Dbicycle_rental