Matt, the pitot covers and as well gear locking pins are stored on the flightdeck. If you play by the rules only the aircraft-paired, one specific set is used to dress the sensors. Exclusively.
There is a stowage box, you will find a photo in the report. Checking for the covers removed is task of fatal importance.
If you read the report Malaysian Airlines does not regularly carry Pitot Covers in the aircraft, and the Covers were supplied by the support engineer on this occasion from ground stock. This is why I made the comments earlier about having a computerised system recording ground fitting of protective/storage equipment, such as covers, locks and such. The aircraft did not have its own set of Pitot Covers, the crew checking the box, would have no effect. The primary cause of this is the split duties between two engineers, who were not exactly cluey on what exactly each was responsible for, this has caused MANY accidents, change of shifts, sudden change of duty, busy work cycles, etc. The old remember everything in my head and remember to do paper checklists just flies out the window. WRT to the crew this is a case of a non standard local procedure for fitment of these items catching the normal crew out. Still there is no excuse for how they got past 50 knots with red flags on the speed indicators.