Lulos, you may already have known, but the original tailgate is the back door on a station wagon, which could open sideways or down as the owner chose.  I suppose a tailgate party preferred that door down, so as to serve as a table to hold goodies.

I can see "tailgate party" coming from that, but it also is easy for me to imagine that the verb came from the same place, ie "don't tailgate" means "don't get so close to that car's tailgate".  But judging by Glenn's citation the verb may be even older than that.  Maybe the original tailgate was on a pickup truck.

Doh!  I'm feeling a little foolish:  Of course the "original" tailgate wouldn't have been on a station wagon.  Oh, well, sometimes I act as though I'm still 14 and think the world was created only a year or two before I was born.