Is it acceptable to cite people without putting the appropriate diacritics in their names?
I'm not sure how far ethics comes into this, but omitting accents is, effectively, a misspelling of the name. If you need to cite something by Schön, and you instead write "Schon", I'd regard that as analogous to writing "Schöm" or "Schöh": pretty close, but certainly a typo. If I'm reviewing a paper, I will request correction of missing accents as I would for other misspellings, but I wouldn't regard it as an ethical breach.

In practice, there's a bit more leeway for missing accents than for other misspellings, partly due to former limitations in computer typesetting, and partly due the dominance of English as a scientific language and the frequent belief among native English speakers that accents are always optional. As Burak Ulgut says, some authors won't mind missing accents, while others will care a lot -- but since you can't know which is the case for any particular author, you should keep the accents. In some cases, missing an accent may have unfortunate consequences in the author's native language: if you cite a Swedish Dr Hörberg as "Horberg", you've just turned them into "whore mountain".

For citation purposes, the correct spelling is (almost always) the one on the publication itself: if, say, Miloš Blažek chooses to publish as Milos Blazek then that's how you cite him, no matter what it says on his birth certificate.

On the technical side: unfortunately you cannot take it for granted that exported bibliographic records will handle non-ASCII characters correctly, or for that matter that they will handle anything correctly. Bibliography record formats are a mess of partially supported, poorly defined standards interconverted by buggy code, and you should always eyeball the record after importing it to catch any errors (not only accents).

As you say, it's clearly a laborious task, but so are many aspects of writing a thesis :).