I will be arriving at Narita Airport in December and will be staying in Japan for a month, I have two large suitcases because right after Japan I will be going to Vietnam leaving from Haneda Airport. Is there a service that can take one of my luggage from Narita to Haneda and storing it there for approx a month? I've heard there's a limit to the amount of luggage you can bring on a Shinkasen because I will be traveling around Japan and not just staying in one area. Any recommendations?
In general, you will have trouble finding a service that can keep your bags that long. Regular luggage storage at Haneda is for no longer than two weeks, but if you're willing to abuse the system a wee bit, you could use the coin lockers. While you can keep your stuff for a maximum of one week in the locker (link in Japanese), after clearing out the operator holds onto them for 30 days before your goods get disposed of.
I believe most train stations operate a similar system, so you could leave your bags at a convenient midpoint like Tokyo stn or Shinagawa. This would, however, cost you a pretty penny at ¥500 or so per day, which translates to ¥15,000 for a month.
The best solution would likely be to book the same hotel for your arrival and departure from Tokyo, and ask them to hold onto your bags while you travel. They will likely do so for free, or a charge of much less than ¥15,000.
Last but not least, I don't believe there is any official/enforced limit to how many bags you can bring on a Shinkansen. In practice, though, you'll find it very difficult to navigate ticket gates, platforms and the train itself with more than one.