DM's discretion: anything cool should be possible
Like all things, you should wonder if this makes sense in the story you and your players are trying to tell.
What kind of person is Jeff?
Because if the player, playing with Jeff just wants to abuse the mechanic of full cover, this isn't really telling a compelling story.
If Jeff has a background in war where he saw 80% of his fighting companions slowly being picked of by volleys of arrows and he still has nightmares about it.
Now this could make some fun story telling.
How to create it
Jeff either needs to fix the leather straps to the door too. Achieving is quite difficult, as these straps should hold the weight of the door without hurting your hands/arms like crazy.
So that would require a hard DC leather working INT check to make that happen. You can add more depth to allow any score to be rolled and the difference between DC 20 and the score is the amount of ours the leather straps hold and they simply break, one after the other after such time.
I chose INT ability as you need to understand how the mechanics work and make sure the leather will stay fixed
Type of wooden door, statwise
Yet gaining full cover (and thus immune for targeted attacks) should come at a hefty price.
So for the fun of it I checked on the densities of wood.
So presuming the door is solid and roughly has the size of 3ft x 7 ft and 4 inches thick. Depending on the type of wood, it should weigh between 140 lbs and 280 lbs.
An solid Oaken door would weigh 280 lbs and I would reckon about 4d10 HP.
A solid Cedar door would weigh 140 lbs and I would give it abot 2d10 HP.
So you can see that carrying some gear together with the door would become cumbersome, like it should be.
Pushing a door around would make it flip over, so not very practical.
My personal take on wooden shield-doors
This is just spit-balling to get toward something that might work at a price but yet gives significant advantage.
I would allow it with the following categories:
2d10 (11) HP, AC 14 and you move as if Encumbered, weighs 140 lbs and has DC 15 to attach leather straps to it.
4d10 (22) HP, AC 16 and you move as if Heavily Encumbered weighs 280 lbs and has DC 20 to attach leather straps to it.
The reasons for encumbrance are the weight and the difficulty to effectively move with the door and staying in it's cover.
Being encumbered by the weight on to of carrying this wooden shield stacks an extra penalty of 10 ft to you speed.
I would give Wooden shield doors:
- immunity to Poison and Psychic
- resistance to Piercing and Cold
- vulnerability to Acid, Fire, Necrotic and Radiant
And have the Two-handed and Heavy properties.