I have 30/06 rifle with a serial number H6872 I am wanting to know the date and manufacturer of this gun

Most will say like, Remington, Like their 30.06 SemiAuto WoodsMaster, Very Nice.

If it's a remington then you might want to look here.

Remington Society of America - questions/barrelcodes

What make is it?

You need to provide the make and model, serial numbers are not exclusive to make and model.

My M40 just turned 47 yrs old and with the exception of the wooden stock which has been changed around 4 or 5 times, the addition of state of the art optics/range finder... it's as deadly today as it was when I first zero'd her!

Semper Fi! 8541 - 2/5