About converting, I'm not sure. Some procedures, like Moderna's or Inovio's, will likely need totally different manufacturing processes. I don't know about Oxford's (which uses an adeno-associated virus).
It looks like a non-trivial effort to do.
Rather than converting, most of the companies involved are ramping up manufacturing at-risk during the trials and some, like Vaccitech / Jenner Institute, are actively seeking partnerships to increase manufacturing capacity already.
Oxford is making an "adenovirus vector vaccine" Effectively they are using a adenovirus with some DNA inside it that will make the spike protein of the coronavirus which the immune system will then attack. No eggs involved.
The egg-based vaccines we have work by taking a human virus, letting it evolve in chicken embryos until it is more adapted to chickens and poorly adapted to humans, and then using that weakened virus as the starting point to make a vaccine.
Interesting overview of covid-vaccines here
https://blogs.sciencemag.org/pipeline/archives/2020/04/23/a-...
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But I guess my question remains: Can we scale up the manufacturing, packaging and distribution using existing (private) infra, or do we need do build this stuff from scratch?
That's why I'm hoping for multiple vaccines to succeed. Like that there will be a faster distribution to the population.
Whether we'll get there or not, we'll have to see. The immune system can act worse than the most spoiled of children.