To my knowledge, yes if playing normally from the discs, no if you rip your games by SD or USB and use DIOS MIOS or Devolution (almost definitely Nintendon't as well, but I've never used it) as a loader. On DIOS MIOS, the feature to save GC saves to SD or USB is called No More Memory or NMM and must be turned on. Devolution's preferences just explicitly ask if you want to emulate the memory card or not, and lets you choose to have a shared virtual card or per-game virtual cards. This seems to work perfectly in Devolution, but DIOS MIOS has occasional compatibility issues with games like Smash Bros. Melee so you would need a physical GC card in DIOS MIOS for that one or the game locks up.
You can use alternate controllers either if you have the physical GC discs for verification purposes and you're using Devolution (in which case, the Wii can connect via Bluetooth to use a Wii Classic Controller [Pro], a DualShock 3, or a Wii U Pro Controller as a GC controller), or if you're using Nintendon't (which can use USB controllers as GC controllers) and can live with its smaller compatibility list. Using a disc or DIOS MIOS, you're limited to the GC controller ports (if present).
Also keep in mind that if you have a Wii with no GC ports or memory slots, you won't be able to rip or verify the GC discs for Devolution, so you'd need a buddy with a hacked Wii with GC compatibility, and you'd need to sync one of your own Wii Remotes to their Wii and use it to perform the Devolution disc verification process there, which will save to the active remote, and allow you to go back home and THEN run the ISO in Devolution with the same remote and Devolution will copy the file off of the remote.
(Yes, often forgotten, Wii Remotes have a tiny bit of onboard file storage, used officially for bringing your Miis to friends' houses)