Without seeing the actual unit, I doubt the display 'lights' are driven by the chip.
That MM58342 (VFD driver) drives the display segments of the VFD display tube.
The low 5V rail voltage will most likely be due to something loading down the supply, (expect to find something warmer than usual in the PSU circuit), or the PSU itself is being 'told' to deliver less power than is specc'ed - hence the dim illumination.
The electronics and display probably have a reasonable tolerance to voltage droop - so they operate normally - at least outwardly. If you can, see if you can 'disconnect' the 5V supply to individual parts of the circuit - to see if any single one is pulling the rail down.
Or equally possible - the input side to the 5V regulator is sagging - so the regulator can't actually get up to the nominal 5V potential - that dirty-side voltage should probably be something like 9-15V or more if other rails are also derived from it.
Of course... if the 'lights' you refer to are the actual VFD character segments - then we are chasing a completely different problem (possibly the VFD tube itself, or the HV supply), but the low 5V rail is certianly worth chasing down.