If it worked correctly before now then something has changed. Check for good electrical connections at the plug and at the cam. If you have it a tiny spray of electric contact cleaner may help- be sure it evaporates before powering. If you don't have that then plugging and unplugging the connections several times may help; no extra force needed for this.
To verify that the cam is not malfunctioning take it inside and try powering it with a phone charger. If it works reliably doing that several times then your problem will be somewhere in the power supply in the car. To test that, unplug the cam, start the car, then plug the cam in. If the cam then starts and boots properly when tried several times the trouble will probably be on the 12V side of things. If your car battery is near it's end you will have to replace it soon anyway so I would go ahead and do that after all this.
If the new battery doesn't solve the problem then try a car phone charger. If that works then the cam PS has gone bad. Normally they either work or do not work, but sometimes they fail slowly giving this result. It is very unlikely that the sd card is causing this but now try a known good one instead. If you are still stuck then resetting the cam or reflashing the firmware are all that is left to try.
Phil