Pack 'em in cardboard boxes of suitable sizes. Ensure that they are fully packed so there is not internal shifting. Fill voids with bubble wrap, newspaper, whatever. Then build WOOD shipping boxes to fit. Add cardboard or other material between your c. box and your wood boxes for cushioning.
wood boxes: 2 x 4 lumber frames and cross pieces, and 1/2 inch plywood.
I built such boxes years ago (most of them much bigger) to ship a household via cargo ship from US to Venezuela, worked great. We overbuilt another one to safely ship a quality upright piano too.
That's the old school way I guess. Ask your local Mailbox USA or kinko's/fed ex or whatever for their recommendations.

Though it would be impractical with a large number of records, shipping the LPs outside of the sleeves is a collector's standard, as it prevents seam splits, where the vinyl slices through the LP jacket during shipping--you might wish to do so with your rarest albums.

What do you put around the LP to protect it in shipping if you take it out of its sleeve?

How do labels ship large numbers of LPs? There has got to be a standard method.

Perhaps, I was unclear. The most basic way would be to leave the disc in the original inner sleeve, and just prevent the outer sleeve/jacket from potential seam splitting. The truly anal among collectors, would place the LP in a separate sleeve (maybe one of those rice lined jobs or plastic lined ones) and put the original inner sleeve inside the jacket. Those might go into a clear polypropylene LP sleeve and everything into a heavy mailer. I'm not actually fond of record mailers, because they don't provide enough "crush room" to prevent bumped corners if the package is dropped. A large box for a single LP is preferable when shipping an individual album, but, good luck getting a dealer to do so. I've had $100+ LPs ruined because of seam splits during transit.

I'm pretty sure labels usually just ship LPs in the cardboard boxes they come in from the pressing plant (probably a 25 count).

Quote:
I'm pretty sure labels usually just ship LPs in the cardboard boxes they come in from the pressing plant (probably a 25 count).

Well, yes and no. I worked for a record press and our customers would typically request either 25-count or 50-count boxes (lower counts for gatefold sleeves, of course). Then there was the Franklin Mint, which repackaged everything into single box-set mailers in PA, so those we shipped in 150-count boxes, which were shrink-wrapped and skidded -- individually, those large boxes were too unstable to ship.

Individual promo copies were boxed in 14" X 14" mailers, with cardboard spacers -- and despite audiophile legend, "white label" promos were not necessarily better pressings. They were typically done first, which guaranteeded fresh stampers, but frequently were pressed before the vinyl achieved maximum flow, so they frequently had underfill problems in the outer grooves.

Record presses considered promo copies, which required us to stop the presses to load up special labels, a pain in the butt. They required extra work, which meant less profit per press, so very few presses took the care with promos that we took with production copies where we could make real money (frequently, a whopping 17 cents per disc).

I was hoping you might chime in with some firsthand knowledge with regard to how many per box. That's interesting about the WLP copies often being underfilled.