So the best odds to find this statistic is to look at the games that ended in a 0-0 tie or went into the 4th quarter without scoring. This way you know that the either team hasn't scored for a good majority of the game, meaning that the odds are much better that there have bee numerous interceptions before the first score.
There have been 73 games that have ended in a 0-0 tie, and the last one to happen was in 1943 with the New York Giants and the Detroit Lions. I observed a list of all NFL games that ended in a tie game, but the team stats for those games are only held back until 1935. There were two games that had 7 interceptions in them, but the game still ended in a tie. October 1 1939, Washington Redskins vs. New York Giants and the other was the Detroit Lions vs. Chicago Cardinals on September 15, 1940. The way the NFL's offense used to be run is not as dominant as it is now. Tie games in the 40s were very prevalent. But, once the formation of the offensive line changed then the tie games became very scarce.
I also researched games that games that went late into the game before scoring. November 27 2007, the Steelers beat the Dolphins 3-0 by a field goal with 17 seconds left. Although, there was only one interception in this game. The low scoring games I looked at didn't nearly have 7 interceptions, but that game went scoreless.
This link below (I'm not sure if you already looked this website up) is a vey good site for finding NFL statistics. It has statistics dating back to the 1920's.