What is the MLB record for most errors by one team in one season during the modern era?
Sporting News record book suggests:
867 by Washington in 1886 (122 games)
If you only want after 1900 then
425 by Detroit in 1901 (136 games)
According to baseball-reference.com, since 2000, the leagues (with 30 teams) have never recorded more than 3500 errors. Between 1901 and 1917 the leagues (with only 16 teams) never recorded less than 3500 errors.
For more information, root around on baseball-reference.com. I don't know of a perfect summary, but you can see team fielding stats for every individual year. As an example, to see team stats for 2014, go to http://www.baseball-reference.com/leagues/MLB/2014-standard-fielding.shtml