You may want to take your child to a paediatric hospital (Kinderkliniken) because regular hospitals often don't have separate paediatric facilities. The main difference among Kinderkliniken is that some have surgery facilities and some don't which could be a consideration if your child has a broken bone or appendicitis. Be aware that you may or may not find an English speaker at the following locations.
Vienna has its own Children's Hospital, the St Anna Kinderspital, located in the 9th District fairly close to the AKH. St Anna Kinderspital has an Ambulanz (Accident & Emergency) that is open 24 hours a day, however serious injuries and illnesses will be refered to Vienna's main hopsital the Allgemeines Krankenhaus (AKH).
Allgemeines Krankenhaus (AKH)
1090, Wahringer Gurtel 18-20
Tel: 01 404 00-0
Ambulanz: 01 404 00 1964
Gottfried Von Preyer'sches Kinderspital
1100, Schrankenbergassse 31
Tel: 601 13
Donauspital Im SMZ Ost Journaldienst Für Notfälle
1220, Langobardenstrasse 122
Tel: 288 02
Mautner-Markhofsches Kinderspital
1030, Juchgasse 25
Tel: 711 65
St Anna Kinderspital
1090, Kinderspitalgasse 6
Tel: 01 40 170-0
Ambulanz: 01 40 170 2100
Kinderklinik Glanzing im Wilhelminenspital
1160, Montleartstrasse 37
Tel: 49 150 0