No reason it would not work, but AFAIK there are no barrels for the M14/M1A in that caliber. Douglas makes M1 barrels for 7mm-08, so they might make an M14 barrel in that caliber. You can call them or Brownells and see what is available. Note that Douglas barrels for the M14/M1A in 7.62 run $250-290.
Unless you can find a barrel maker who makes a 7mm-08 barrel for the M14/M1A, it would be a gunsmith proposition.
That means a gunsmith would have to make the barrel from a blank, a pretty difficult (and expensive) proposition with the threads and gas port, not to mention the rear shaping and cuts for the handguard clip. The port size would be somewhat trial and error. I would think the barrel tang would have to be threaded, then the barrel installed and headspaced, then removed and the other threads cut, the rear shaping done, the gas port drilled and the spline cuts made.
You can try to get some quotes from experienced Garand/M14 gunsmiths, but I don't think you will like the bottom line, which I think will be a grand or more.
(Every time I write that something like this is difficult and expensive, I hear from some guy who says his brother-in-law will do it for $2.50 and a case of beer. I eagerly await a similar response, but think this one might require two cases of beer.)
Jim