When we are telling a story, we usually use either the present tense or the past tense. (http://en.wikipedia.org/wiki/Narrative_mode) Your first sentence uses the present tense, which is fine:
There is [present existence] this one driver, who goes and gets [present habit] a coffee on the way and you have [present habit] to wait like for 10 minutes.
Your second sentence mixes the present tense and past tense, which is not fine:
There is [present existence] this one driver, who would [past habit] go and get a coffee on the way, and you would [past habit] have to wait like for 10 minutes.
So just change is to was:
There was [past existence] this one driver, who would [past habit] go and get a coffee on the way, and you would [past habit] have to wait like for 10 minutes.
(You also omitted the to from have to in your second sentence. I assume that's just a slip.)