I ain't great at math, but I'm passable at armchair 'research' via resources. From a web search, Wikipedia says the 183rd day's noon is the midpoint of the year, and in leap years, it's the 184th day's midnight. Each of these is apparently consistently July 2.
So according to Almanac.com finding July 2's day of the week for a given year in the 21st century is a "simple" algorithm: remainder of (((3rdYearDigit+4thYearDigit)×1.25+DayOfMonth-1)/7). Brute forcing it through a spreadsheet shows me that there are 15 wednesdays as July 2 this century. That's just over 1/7 so looks to be slightly more than every 7 years, on average. That's average, though: the next ones are in 2031 (6y), then 2036 (5y), then 2042 (6y), then 2053 (11y), for example. So it's kinda common, but there can be a long 'wait' for it too.