if they are not very white, it is usually because they have been washed with non-white items.
"nearly white" or "mostly white" will not do.
If he wears white shirts for work, cotton ones will stand up to higher temp washes than polycotton, and will look whiter, but watch out for the collars wrinkling at high temperatures as the interlining will be synthetic. White cotton T shirts or vests etc can go in with them, but not polycotton sheets if you need a hot wash.
You need to use a powder, not a liquid, for dazzling whites.
If he has sweaty pits, a white cotton Tee will protect the shirt and also improve the appearance of whiteness.
I have generally been pleased with white cotton shirts from Debenhams. They almost always have a sale on, and if you have a store card they will bombard you with offers and vouchers. You will probably get free delivery too. Some of their shirts come in packs of three at a reduced price. On-line ordering has better stock than your local store, though you may find local clearance reductions. Easy-iron are. Some cotton shirts feel soft when new, this is due to the thread being lower quality, they will never iron crisp. Run a tape measure over him to see if he suits tailored, slim, regular or porky, and if he has long or short arms. there is a fitting guide online.
I am noted for the brilliance of my office shirts. I have been laundring my own shirts since I was 13 and I think any man should be able to wash and iron his own to the standard he requires.