Does the bold part mean "the paper sizes vary based on the business, and it also varies based on the operational requirements of a company, but it's degree of variance is lesser than the one based on the business"?
Yes, that's exactly what it means.
I would have punctuated the original sentence as:
Workers may prefer a tray that matches with paper sizes, which vary based on the business and, to a lesser degree, the operational requirements of a company.
where the commas make the phrase "to a lesser degree" operate as a sort of parenthetical, and thus make the sentence easier to parse.
You can see clearly that there are two reasons why the workers' preferences might vary: business concerns and operational requirements. Then, from the parenthetical "to a lesser degree", you can tell that the second concern (operational requirements) is less important than the first concern (business).
The phrase "to a lesser degree" is a common idiom in the English language. The adjective "lesser" could also be substituted for "greater" to invert the meaning. You also commonly see "extent" in place of "degree"; this has essentially the same meaning.
You can read "to a lesser degree/extent" as "less than"; "to a greater degree/extent" can be read as "greater than" or "more".