You don't need to have 10,000 caps on your person at any given time as this is entirely accumulative.
It is worth noting, that items sold in your personal vendor at your C.A.M.P and transferring caps between friends do NOT count towards this total. It has to be earned from events, through the games vendors, mission rewards and the season board unlocks.
The easiest way I found was to work through the station vendors and sell as much to deplete the daily vendor wallet. About 1,000 caps a day.
I would do Uranium Fever, take all the meat and weapons, cook all the glowing meat at WhiteSprings station and sell the cooked meat to the vendor and store any extra in stash for the following day.
Jumping through events on the map, will also net you a couple hundred caps a time so it's best to stick to the popular ones like Uranium, Scorched Queen, Daily Ops etc.