No. It is your choice which cremation service you use.
You must ensure you have clear descriptions of each stage of the service from collection to the final return, scattering or burial of the ashes and be sure you understand and are assured by them. If your vet is unable to provide this then do your own research.
Unfortunately most of the companies dealing with veterinary surgeries collect the waste at the same time and this is often a main factor in a surgery choosing a company. With large numbers of surgeries to collect from the handling is often more in line with a waste collection than a cremation service. Although the crematorium may look very nice on their website, the service they provide through the surgeries may be quite different to the one they describe and provide if you go direct to them.
So it is always best to take control and do your own research. The inspection system of the APPCC is the most rigorous available and ensures you get the service that is described to you, handled with the utmost respect. Look out for the “Independently Inspected and Approved” APPCC logo. Other members may not have been inspected but all have agreed to abide by the Code of Practice.