Carbon Recapture projects give Diplomatic Favor, not Diplomatic Victory Points.
But you can get 7 DVP from wonders and 2 from tech, which leaves only 11 others to find. That's not so many, I think. My latest Diplomatic Victory was turn 215 and it would have been sooner if I hadn't accidentally voted for the wrong thing because I wasn't paying attention.
I checked my last three games (Immortal, because I'm lazy lately) and they entered the Medieval Era on turns 96, 94, and 96. So, let's say turn 95. The Congress votes every 30 turns. So, you'll get to vote on turns 95, 125, 155, 185, and 215. Even if you only average 1 point from each, which is usually quite easy to do, that's another 6 points added to the 9 that you got from wonders and tech, leaving you with just 5 more. If you win the +2 vote even once, then you only need to find 3 more points from aid requests, contests, or other votes.
Honestly, with the latest changes, Diplomatic Victory is probably faster and easier than Scientific or Cultural Victory.