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. The same subsection allows a joint permit, owner and contractor together, "if the contractor is 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 when a pour needs a pump it is our own trailer line pump on 2.5 and 3 inch hose. Ask the VDOT district or residency office serving your county whose name belongs on the permit before the concrete is ordered.
Where the entrance stops and the driveway starts
The permit is for the connection inside the right of way, and 24VAC30-151-40(A) makes a permit valid only there and on VDOT-owned property. Its conditions can still reach sight distance, though: under 24VAC30-73-90(A) the district administrator's designee "may require the property owner to grade slopes, clear brush, remove trees, or conduct other similar efforts necessary to provide the safest possible means of ingress and egress that can be reasonably achieved." Back on your own land the driveway is a local question rather than VDOT's, so ask the county, and the town if the lot sits inside one, what it requires there.
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. LUP-PE is VDOT's form name, NCDOT's form is titled a street and driveway access permit application, and for a house the North Carolina rule starts from the opposite default: under 19A NCAC 02B .0601(a), driveway connections to residences are "normally excluded" from its permit requirement, though they "may be included at the option of the Department" in the circumstances the rule lists. That is why the Virginia paragraph is the wrong paragraph to carry south. The NCDOT side is written separately for exactly that reason.