What the letters mean, and what kind of paper this is
LUP is VDOT's land use permit, the family of forms that gives written permission for somebody other than VDOT to do something inside a state highway right of way. PE is the private entrance member of that family. So this is not a driveway approval in the ordinary sense of the phrase. It is permission granted against the road, not against your lot, and that is the whole reason it exists as a separate instrument with a separate name.
Private entrance is a defined term rather than a description. 24VAC30-73-10 draws the category by what the connection serves, and it sits beside two other defined entrance categories in the same section, so a drive that carries more houses than the definition allows is a different permit with different requirements rather than a larger version of this one. Which category your address lands in, which road classes the form reaches, and what VDOT charges for it are laid out on our VDOT entrance permit page. This entry stops at what the words mean.
Whose name goes on it
Under 24VAC30-151-40(A) an entrance permit runs to the adjacent property owner. A contractor can appear on the same permit, but jointly with that owner and only as the owner's authorized agent, which is a role the owner grants rather than one the trade confers.
So the default sits with the property owner, and being the crew that places the concrete does not make anybody the permittee. We pour driveways, and the concrete goes in through our own trailer line pump on 2.5 and 3 inch hose. We do not hold your entrance permit and we will never tell you we cleared it.
Where the entrance stops and the driveway starts
The permit reaches the connection inside the right of way and stops there, because nothing on your own land is VDOT's to permit in the first place. What governs back there is the local building official, and the zoning administrator as well in a locality that has zoning. One instrument, one strip of ground.
Even inside the strip the state standard is not automatically the last word. 24VAC30-73-60(C) provides that entrance standards established by localities that are stricter than VDOT's shall govern, so where a locality's entrance standard is stricter than VDOT's, the locality's standard governs.
Two things the name does not carry
It is not a clearance. 24VAC30-151-20 grants the permittee only the rights the permit itself sets out, and only so far as VDOT's own rights in that ground reach, and it puts the job of getting permission from anyone else with an interest in the property on the permittee. Whatever else is recorded across that strip, or buried under it, remains its own conversation with its own holder.
And the acronym stops at the state line. There is no LUP-PE in North Carolina and no comparable acronym, since NCDOT's form is titled a street and driveway access permit application, and the rule there for a connection serving a house runs on the opposite default from Virginia's, which is why the Virginia paragraph is the wrong paragraph to carry south. The NCDOT side is written separately for exactly that reason.