We do it ourselves in our rescue, but we get vaccines from Online stores (Dr Foster and Smith, Jeffers Pet) and when we get them they have ice packs, so are cold, we put them directly in the fridge.
Also it takes some skill and learning to do it correctly, and the puppy will move, yes, sometimes they don't care but even so must be held firmly.
It's often better to visit the vet with your individual puppy, so your vet can view puppy, listen to heart, lungs, etc., and do stool samples and issue dewormers.
If you are doing it to save money I recommend against it. If you're doing it because you're that knowledgeable about puppies and can detect an illness, then by all means do it.
But only use 5-ways. DO NOT Mess with 7-ways at home.
Community Practice Vaccination Protocols from the College of Veterinary Medicine
Puppies 6,10 and 14 weeks or 8, 12 and 16 weeks
Repeat 1 year later
Then repeat every 3 years
Each patient should be evaluated for increased risk of exposure – may consider yearly protocols for these patients
Give in right shoulder