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When I read the following sentence,
少しでも長生きをという願いをかなえるために、医療は目覚ましい進歩を遂げてきた。
I think there is a missing verb as indicated as follows.
少しでも長生きを [missing verb] という願いをかなえるために、医療は目覚ましい進歩を遂げてきた。
Why can を be followed by non-verb? What kind of grammar is being applied here?
As you have correctly guessed, this と is the quotative particle and the part before it is a quote.
「少しでも長生きを」という願いをかなえるために、医療は目覚ましい進歩を遂げてきた。
A sentence that ends with を is commonly used as an expression which means "We/I/You need/want ～", "Give us ～", "Accomplish ～", etc., depending on the context.
One may say that a verb したい is omitted in this particular case, because 長生き is usually used with する:
「少しでも長生きを（したい）」という願いを…
But I don't think people bother to imagine the omitted verb in a case like this. Perhaps what you have to understand is that the noun mentioned before を is strongly desired.
