Every player gets 0.1 capture points for every second they push the cart. This does not have to be continuous, or even in the same life, but it does have to be unblocked. It doesn't matter what class you are, what your capture multiplier is, or how many other players are on the cart. Only once your amount of capture points reaches the next whole number does it count, so in essence you get 1 capture point every 10 seconds.
However, this is a mapper-defined value ("partial capture rate"); mapmakers can decide to raise or lower this rate, so some maps can have faster or slower rates of giving out capture points. On most official payload maps this value is 0.1 (hence 10 seconds being the conventional wisdom), but a few are different - for example, it's 0.02 on Upward, so there you have to push the cart for a full 50 seconds to get one capture point.