I don't think some of the schools have the technology yet. - - OK?

The sentence is grammatical as it stands, but, as you say, a native speaker would probably not formulate it in that way. I think it would work better if you replace 'some' with 'many' (if the shortfall is substantial) or 'all', or else you might consider recasting the sentence in this way: I think (i.e. my assessment is that) some of the schools don't have the technology yet.

OK, thank you!

What if the speaker wants to express the amount that is described by 'some'.

It is grammatically feasible and it is sensible for a given situation.

Just my writing idiom, but I would prefer

"I think some of the schools don't have the technology yet."

If I wanted to say "all" I'd say

"I think the schools don't have the technology yet."

or even

"I think some of the schools don't yet have the technology."

Etc.

They all seem to imply that the schools (some, most, all) don't have the technology now, but it's on it's way.

Much of the meaning also depends on which word in the sentence is emphasized when speaking.

I don't think all of the schools .....

I don't think/believe that some of the schools have the technology yet.