I give my puppies the initial series (to 16 weeks) of DHLPP vaccinations at home. I just buy them at tractor supply company. I think it's a 7 way vaccine or something - just look at the boxes, they'll tell you which one to start with.
HOWEVER, it is VERY important that a-) you keep them cold/refrigerated and be aware that if at any point int he process of shipping and storage they are not kept chilled they are likely to become ineffective and will not work b-) follow directions on how often and which and c-) still take your puppy to the vet. And that you're going to have to give the rabies via vet, at least.
Also very important to know that places like your vet, puppy classes, airlines etc. will not accept those vaccinations as 'valid' for purposes of attending classes, boarding, etc. To do those you'll have to either get titers taken at the vet or do the whole series again at the vet. So it's not actually money saving.
I do Distemper+Parvo (in the combo vaccination) myself. It's an easy subcutaneous injection, and my dogs are trained to be still and calm while I administer.
By law, only a vet can administer the Rabies vaccination.
But honestly, if you have any doubts about the how-to or how it shows up on record, I would recommend seeking advice from a vet.
Like others have said, it can be done, but if you ever plan on boarding, grooming, or doing classes, I would just have the vet do it. I don't think any of those establishments will accept your Tractor Supply or other farm store receipt as proof of vaccination.