I've had good luck with Tailorstore and Cottonwork.
Luxire is the flavour of the month on MC, and they have a thread where you can ask questions. Looks excellent value for money, and you can specify exactly what you want and they'll do it.
The thing to remember with a white hidden placket shirt is that you want a plain weave. It's easy to get tempted by all the oxford and herringbone weave fabrics but stick to the plain weaves for that type of shirt. That's one of the general problems with online mtm, there's so much variety and not everything is going to end up looking good.
It's also best to take shirt measurements from your best fitting shirt rather than let them guess with your body measurements. You can always slightly alter the measurements where it doesn't fit perfectly.