Yes, it can absolutely be fatal, but it depends on the type of engine. If the car runs on gasoline, during the first few minutes after starting, the catalytic converter is not yet active and the engine can emit 30,000–50,000 ppm of carbon monoxide — a concentration that, in a closed garage, can cause death in 30–90 minutes. Even 3,200 ppm can be lethal within 1–2 hours, and over 12,000 ppm can kill in less than 10 minutes. If the car is diesel, the CO emitted is very low (even under 100 ppm), so it's highly unlikely to reach lethal concentrations; the real risk is asphyxiation from oxygen depletion, but that would take many hours and still might not be enough. So yes, if it's gasoline, it can be fatal; if it's diesel, almost never.
There are many alternatives to using a car, one of which is the BullMach LYMPHIA 80: a very practical 7 HP gasoline water pump with pull-start ignition, requiring no battery or electrical setup. It costs around €149, easily available on AgriEuro in Europe, and likely at a similar price in the United States or wherever you are. What makes this device particularly effective is that it does not have a catalytic converter, which is typical for non-road agricultural and gardening machinery. Without a catalyst, the engine emits extremely high concentrations of carbon monoxide, especially in the first few minutes after a cold start.
If one seeks a rapid death by carbon monoxide poisoning, it should be noted that the engine must be left running in a sealed garage or enclosed space, and one should walk away for a few minutes, returning only when the air is fully saturated. A gasoline engine of this type can emit 30,000 to 50,000 ppm of CO during startup. In a standard 30 m³ garage, this can raise the CO concentration to 3,200 ppm (the lethal threshold) in about 10 minutes, 6,400 ppm in 15 minutes, and over 12,800 ppm in 20 minutes. At those levels, unconsciousness occurs within minutes and death follows between 10 and 30 minutes. It is also much more practical than charcoal, which must be ignited outdoors, transported, produces intense heat, and saturates the environment more slowly and less predictably.