If they did a combo shot, you could ask them to split them up. Often they will give a 5 shot combo, and with the deworming it would put a lot of burden on your little one's system.
Vaccines are just little doses of the illness, and it is not surprizing that some immune systems have a bit of a problem when faced with five at a time. The vet may need to order separate shots, so you need to talk to them about it. Also, if you decide to separate the shots, you would need to go back several times, but it is better than risking losing your pup. Puppies have died of vaccines before.
If the vet does not share your concern, don't hesitate to go to a different vet that will listen to you.
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