Is it possible to complete the City Influence Quests before completing the main Thieves' Guild quest line? If so, how do I access these quests?

It should be possible. You need to start the main guild quest line but once become member, start doing those miniquests for Delvin and/or Vex. The reputation quests then should become available when you complete five miniquests in each four other main hold capitals.

I do it every time.

I've never heard of the city influence quests. Are you talking about the ones where the two people in the thieves guild has you doing misc quests like forging ledgers, robbing specific items from houses and stuff like that?

No, those forging ledgers, robbing specific items are those miniquests I referred. Reputation quests however are bigger ones and there are four of in total, one in each other main hold capitals.

This. Keep notes or you will be repeating jobs.