The amphibious invasion order has "issues". Move the fleet adjacent to where you want to invade, then move the troops ashore. They should receive supplies upon landing. I'm assuming that you haven't turned off supply to the units, and have sufficient supplies in your stockpile and reaching the port which you're using as a launch point. The Supply overlay should show you whether you have enough supplies available at the port.
Several of the previous posts might be of help if you're sending a massive fleet and landing party, but not very helpful if you're only sending a couple of divisions and a handful of ships, but still not getting supplies.
As for invading Japan, your basic Transports don't have the range to reach Japan from Mexico directly (except for rebasing to a friendly port), so you either need the more advanced Landing Craft, or else grab an island somewhere along the way as a forward base. Note that the AI gets to cheat, and can launch an invasion from any distance without supply problems.