To my knowledge, Remington has never offered the 338LM as a factory chamber.
That being said their custom shop may very well offer it.
Many, many gunsmiths have been performing the conversion for a number of years. I myself have done a couple. The first being a 338 Ackley Improved on a Left handed 700 action belonging to an old friend of mine. It was a winter "frankenboomer" project that we did years ago.
Essentially what you need to do is find a magnum size 700 and then ship it off to someone who understands "boomer class" magnum rifles. Then stand by to liquidate some of your disposable income. If your going to go through the effort of revamping the receiver to accommodate this cartridge it would be well worth the time/effort to have the receiver blueprinted as well since the conversion already involves a fair amount of work. Also undestand that once all this work is done its basically a one way street. It's not going to be an easy accomplishment to backtrack and set the action up for a more mainstream magnum cartridge. The 338LM is a great cartridge but it is not one of the more affordable ones. I own two, one in a 30 cal wildcat and one in the parent cartridge set up for 300 grain bullets. Neither of which are cheap to shoot. Barrel life is limited to less than 1000 rounds on a the 30 cal and the other is right at about 1300-1500. The ammunition commercially is well over $100.00 for 20 rounds, if you handload that backs off a bit, but the initial investment in components and tooling is high. Brass typically runs between $1.75 and $2.50 a piece depending on source and availability. 105 grains of powder at a time eats a lb pretty quickly.
Magazines: The BEST magazine available for this cartridge is the one offered by Sako for the TRG. Use this and the gun will feed as fast as you can manipulate the bolt. They are not cheap though at over $170 bucks a piece. They can also be a bear to get. I make a floor metal/trigger guard assy that uses this part.
FWIW Dakota Arms, Inc. devoted the better part of a year developing a sniper rifle platform for the Jordanian Special operations command in 2004 and 2005. The rifle was eventually called the Scimitar. There are a few of these actions still available to my understanding and they work exceptionally well. Control round feeding, open architecture trigger assy (like a Winchester), 3 position bolt safety, billet bolts, EDM raceways, gun drilled/honed bores, and chromoly receievers. They also have a heavy tang to support long winded/large contour barrels. They also use the TRG magazine.
Pretty sure Remington is offering a 338 Lapua Mag now on their 700 action. I've read a few posts/threads from owners of them off & on for about 2 years now. I think you were overseas when the first model came out. I don't know if it's their standard 700 action with a different bolt face and feed rail - otherwise the same that all the other 700 actions - or one that's using higher strength steel.
Here's one Thread discussing the rifle, including as least on pic of the offering;