Designing vessels that operate above ambient pressure is relatively straightforward since the outwards pressure tends to assist in keeping structure in shape (eg soft drink cans resist crushing better while sealed).
Using hoops stress calculators in theory a one atmosphere pressure needs something like a paint thickness of additional material. Where this gets tricky is resisting pressure inwards, since now any deformation turns the structure from resisting pure compression to resisting bending. Example being an empty soft drink can which will support an adult standing if undamaged, but a slight dent or asymmetry in applied forces will cause it to collapse under a far smaller load.
A guide from CERN gives a rule of thumb on page 7 (section 3.4) of a thickness of 1/100 the chamber diameter when working with steel. For a Falcon 9 fairing 5.2 meters across this gives a thickness of 5.2cm. For the cylindrical part of the fairing 6.6 meters high this gives an area of 107 meters and volume of 5.6 cubic meters. Steel is 7900 kg per cubic meter so total mass is 44292 kg, which is more than the mass of the payload to orbit for a falcon 9, and the 1/100 example provided assumes your chamber is sitting on a foundation rather than being flung through the sky.
Using Carbon fiber would improve this, as would using a partial pressure, but unable to find straightforward examples to math out. For the existing fairings have not been able to find a mass of the sound insulation, but the total fairing mass is around 1900 kgs so even if a properly engineered vacuum fairing is 1/20th calculated above it is still heavier than the current fairing and not yet resisting aerodynamic loading.
The payload fairing has a volume of around 100 cubic meters, so just using helium at normal pressure would reduce the mass/add lift by around 100kg, though this would fall away relatively rapidly as the rocket ascended and fairing bleed down but still 100kg less at launch for around $400 worth of helium seems like it might be useful assuming it did not cause problems for the payload as mentioned in comments.
In summary flying payloads at reduced pressure seems to have a substantial mass penalty - it might be viable if a payload required low pressure/vacuum conditions during launch but would come at a non trivial mass penalty.