After reading up on it, it seems that there are plenty of civilians buried at Arlington, not just military, and that there is a fair amount of leeway in determining who can be laid to rest at the nation's most hallowed cemetery. But I also found that he narrowly avoids outright disqualification to be buried at Arlington.
S.923, passed on June 18th, 1997-
An Act to amend title 38, United States Code, to prohibit interment or memorialization in certain cemeteries of persons committing Federal or State capital crimes.
The vote was 98-0 in favor, and it became law later that year. So Ted Kennedy voted in favor of not allowing convicted felons to be buried at Arlington National Cemetery, among others.
Fortunately, he was only charged with a misdemeanor for his embarrassing actions at Chappaquiddick, and even a grand jury, convened a year later, was only looking at Kennedy for manslaughter and/or perjury, neither of which is a capitol crime. Unless of course you consider whether the only account of the accident (Kennedy's of course) may not have been completely accurate. Here's a good article on that.
So by virtue of having been the brother of JFK and also having spent 40 years moving the country towards socialism, Ted Kennedy will be buried among men and women who gave their very lives for their country.