Can car floor mats really cause accidents?
It is possible for floor mats to jam an accelerator pedal.
Whether this happens or not depends on many factors, including the design of the mats, the design of the pedal, whether the mats are genuine “factory” mats, whether the owner is using the fasteners provided to hold the mats in place, and perhaps some other factors as well.
This type of problem, while publicized in connection with late-model Toyotas, is not new, and even took place 25 or 30 years ago–albeit in isolated instances.
Yes. Floor mats can bunch up and cause the accelerator to stick or, in some cases, keep the brake pedal from operating properly. Combine that will an inexperienced driver and an accident is very possible.
Allegedly. Anything that can potentially get the gas pedal “hung up” can cause an accident, and a floormat certainly has that potential if it doesn;t fit properly or gets “askew”.
Car mats can even prevent accidents:
Toyota will solve the Sequoia rollover problem by placing both floor mats on the passenger’s side! 
The culprit isn’t usually the stock floor mats that came with the car. When a floor mat causes an accident, it is usually one of those heavy duty thick rubber aftermarket floor mats.
I had some cheap ones with weak backings and they would bunch up and interfere with the pedals. I got rid of them.