Can a vet do Wudhu and pray on time while being in a state of impurity due to handling of dogs or does he have to wait to get back home and bathe and pray all the Qadha namaz together?
Touching any Najis item including dogs or its urine etc does need Ghusl at all. You need to wash the part which touched the Najis item with water, then perform Wudhu and pray. It is a sinful act to delay performing the Prayer till it's time goes. Performing the Qadha namaz will not omit the sin of not praying within the time of the prayer.
You must perform your Namaz within its time under any circumstance.
Wassalam.