In a comment for this post, a reader named Alex pointed out that Rakuten runs an English online store that will ship to the US (and other places) and sells baseball cards. It's apparently a collection of shops, so I've found multiple listings for some items. Unlike Ami Ami, they have more than just the most recent cards - I saw a box of the 2001 Upper Deck cards for sale. And they have a promotion going on this coming week - from 7 AM (EST) Monday (June 13) until 1 PM on Tuesday (June 14), they are offering free shipping to the US!
So thanks to Alex for the tip.