Since I started modding the game, I haven't ever played through an entire career. The furthest I got was 1943 when I got my XXI. Shortly after my computer took a dump and I lost everything. In the career, though, I had 600,000+ tons, so I'm guessing you could make it. I used the type VIIC almost exclusively until I got the XXI. What you need to do I try and find a convoy on every patrol. Sink the T2's and T3's. The T3's are especially easy to sink as it usually only takes one shot (without the silver bullet AutoTDC). T2's are usually easy to sink, too. Send a few of those to the bottom on every patrol and you'll reach your goal in no time. I would find it hard not to come across a convoy on every patrol. Sometimes I advoid them just so I don't have to endure the lag.
Lately I have been averaging above 20,000+ tons a patrol in my VIIC. I recently scored 86,000+ tons in a patrol. I'm not sure how many game days my patrol lasts, but I try not to play more than 2 hours at a time seeing as I don't like to save unless something comes up.