In regards to flashing the full width of the deck:
Instead of flashing the entire width of the deck, usually the vinyl would go up the wall several inches. If this is not the case, and there is also no flashing, I would call a vinyl guy (not the same one who is responsible for this atrocity) to come in and do that. they should be able to look after this, heat-welding vinyl-to-vinyl, and use some good sealant at the doorway seam and do a better job and use a better product than most homeowner specials. Below is an image of a this after I reinstalled some of the siding, but you can still see how the vinyl goes up the wall.
In regards to caulking as a solution:
Just to reiterate that there is no cheap way to fix this properly at this point. Whoever built your deck is not someone who should be charging people since they have no clue what they are doing.
You can use caulking for a temporary solution but you should know that it is something you need to check and replace as necessary.
Even then you should keep in mind that even if it looks okay it might be letting a small amount of moisture in that won't show up as a visible leak but can cause rot in your framing over time. I do a lot of repairs where this is the case.
Caulking being relied on for waterproofing usually ends up as an expensive repair at some point.
May be many, many years later, but may also be sooner than you wanted.
I have many photo examples of this, but here is my current project. A roof-top deck where caulking was relied on for sealing flashing seams.
Here is a beam underneath a sliding door at the corner of a house. It had sunk over 1/2" inch and was being held together by
magic and pixie dust™. (the new section is what I added in while replacing.)
I have tons more, but this should be enough to reinforce the point.
Caulking is never a "fix", it is only a band-aid solution that will help delay or prolong the inevitable result of poor workmanship.