I'm not a chemist or immunologist, so I don't know enough to know what alternatives are feasible. But like I mentioned in an earlier post, there are a lot of issues to using inert substances as carrier media, and vaccines contain things like formaldehyde in such trace amounts that the effect of these is concluded to be negligible or significantly less than what one would be exposed to by normal exposure to things like tap water. Which supports a valid case for delaying vaccination if one is concerned about the exposure of a newborn to such substances.
But it's also fair to say that the biggest concern people have about vaccinations is that the immunological response triggered by the live virus, and the assault it has on an immature or otherwise unable to cope immune system. And this response has nothing to do with the other materials that are used in vaccines - instead it is the live virus or viral proteins that triggers this response (exactly as it is designed to do). So ultimately, using alternate materials would not address the larger concerns people have with vaccines.
A point on using animal cells to culture viruses. This may seem like an unsavoury choice, but the alternative is using human cells, which, given the necessary source of human cells for culture, is significantly more unsavoury to many. But it's also important to remember that the vaccine is cultured on animal or human cells. Although the ethical dilemma of using human cells remains consistent regardless, the actual vaccine contains no human (or animal) cells. For example the rubella and chickenpox vaccines used today were originally cultured from the same tissue back in 1962.
Whilst it is fair to suppose that the use of animal tissues could be a problem, there are a number of reasons why it is unlikely to be. Firstly, humans have been exposed to animal transferred diseases since we domesticated them - it is one of the primary sources of human transferable viruses. Secondly, when we use an animal cell for culture we do not use diseased tissue for a start, and biologically it is unfeasible to transfer blood or airborne transferred viruses through cell culture. Finally, using animal cells as culture media does not impart the DNA or RNA of that animal to humans, any more than eating meat alters our DNA.

