The Famicom version supports saving and loading via cassette tape. The Family BASIC keyboard has ports to connect to a tape recorder, and Nintendo released an official one called the Famicom Data Recorder. The western release uses the exact same ROM, but because there is no way to connect the Data Recorder saving and loading don't work.
Later ports remove the save and load option, but the Virtual Console versions (Wii, Wii U, dunno about 3DS) include the features.
The Famicom Disk System version of Vs. Excitebike supports saving and loading with the game's disk.