Should I shovel snow off the vinyl deck

Around here, 4x4s are not uncommon. This table shows that a 4x4 pine post that is 6 ft or shorter can support a load area of up to 10 ft² assuming 50 lbs/ft².

Don't trust me or a random table on the internet; make our own calculations from tables in the building code or other trusted sources and consider having them checked by an engineer.

Based on questions answered in chat you have a case where the contractor didn't measure the deck properly, and built it too high, could not flash properly under the door frame, and performed a "some genius" move putting a transition strip (meant for interior use) over top of the door frame and the vinyl, with caulk under neath.

Stupid.

Two options: 1 cheap and temporary, 1 expensive and proper.

Cheap: Use proper caulk (props to @tester101)

Option 2 - and the door is old so this isn't actually a terrible idea.

Replace the door. In the process, raise the rough opening of the door frame half an inch or so - however much is necessary to be able to put a proper piece of flashing under the door frame and over the vinyl.

You will probably have to custom order a door to fit your new rough opening, but at least you can seal the door area properly.

And props to @bmitch for pointing out that flashing needs to run the entire width of the deck.