I sympathise - I can remember taking baby Barley with me delivering Xmas presents when she was only about 9 weeks - she puked and poo'd but the thing was that the only thing I had available to pop her into for the (reasonably short) trip was our big cool-box ..... I did have to stop and sort of sluice it out ...
When we got home, Simon was due at his afternoon Nursery session, so I quickly bathed Barley in the kitchen sink, wrapped her in a baby blanket, and we went into Nursery with her inside my fleece. Everyone said 'Aaaaaah, isn't she gorgeous - so fluffy and lovely!' (Only me and Si knew what she had looked/smelled like not 20 minutes before .....yeeuch !!)
Lots of puppies suffer from upset tummies when they go to their first real home - like Jane explained very well, this can be due to lots of factors, none of which are your fault. Do keep a watching brief on their liquid intake though, since they need to drink plenty to replace all that water they lose if they are sick or squitty. You can get electrolyte things to put in their water which helps their system balance again, but do consult your Vet to make sure that there is no other underlying problem ...
