Japanese websites for custom cards?

Are there any Japanese websites similar to Shutterfly?
I would like to print custom made front and back cards.
As well as magnets, cups, etc.

Try one of the below sites. I used Vistaprint before when they offered "free" cards (but it comes with ads printed on the back side) for my personal use (not fo business). If you are printing for business, I suggest you opt for higher grade paper stock, as Japanese are quite finicky about business cards.

https://www.vistaprint.jp/?no_redirect=1&xnav=logo
https://raksul.com/businesscard/
https://odahara.jp/

There are lots of others. You can do your own search using the phrase ネット名刺印刷 which means "internet business card printing".