I have a question that is swirling around: I'll be glad of any help in checking my grammar 'logic' to find the least ambiguous solution.

VERSION A
The vaccine specialist company = a company that specialises in vaccines; this is the meaning I want to convey.

Because 'vaccine specialist' describes the Company and it is a compound adjective, it ideally needs a hyphen, as in: The vaccine-specialist company.
I have seen it used without a hyphen.

VERSION B
The specialist vaccine company = a company that has specialist vaccines; NOT what I want to convey.
Because 'specialist' appears to describe the vaccine and 'specialist vaccine' is a compound adjective, it ideally needs a hyphen, as in: The specialist-vaccine company.
Again,to me, this means the vaccines (and not the company) are 'specialist'.

Any polite, useful and bright ideas?
Thank you.

You're right on all accounts.

To portray the idea of "a company that specialises in vaccines", you could use
vaccine-specialist company. To make sure I was understood though, I'd go for vaccine specialists, if possible.

More on compound adjectives here.