Need a quick fix for leaking door frame. During flood
IF you can drive during this event  then stuff something in it  for now and go get sandbags, or supplies for making them if they are available.  If you can NOT drive, there may assistance available from your local municipality to have officials or  volunteers bring you sandbags.
You can apply caulking to the gap  under the threshold on the exterior, (this should have been done at the time the door was installed) this will only keep the water out if the water is not higher then the threshold. Most caulking can not be applied on a wet surface,I have read that Lexel by Sashco can be applied in/on wet surfaces.
Doors are not designed to stop water intrusion from flooding caused by tropical storms. They may stop driving rain but if the water outside is not flowing away and is accumulating faster then it can be absorbed into the ground, flooding, then it will make its way in where ever it can and door are perfect for that. Typical  entry doors will not stop that amount of water unless they are designed to do so, such as a ship or submarine door.
The usual way of keeping water out of houses during extreme weather events is to build a structure that stops the water from reaching the structure or diverting it away from the structure. (Or building the house above the high water mark, such as on pilings)  Sandbag walls are used in emergency's or in places where permanent structures are not in place.
Sandbags are not 100% effective at stopping all water but they will reduce the impact of flooding.
In some locations on the planet these events warrant a permanent structure to deal with the repeated events. In others it may be a &quot;hundred year event&quot; or an anomaly and sandbags or other temporary solutions are applied.  As the planets weather changes, the frequency of these events can increase.
In your case you need to stop the water from reaching the door now, not modify the door. Stuffing something in the gap under the threshold may help to keep the intrusion to a  minimum.
A sandbag wall is the best solution until the storm passes. Then you can determine if it is prudent to plan for future events and how best to do so.
Sandbagging Demonstration.