Were you using MultiBit "Classic" or MultiBit HD?
If you were using Classic, and CoinCorner supports "importing" or "sweeping" Bitcoin Private Keys, then you should be able to that, as MultiBit Classic allows you to export private keys. However, with MultiBit HD, you will only have the "Seed Mnemonic" (aka wallet words)... so unless CoinCorner supports restoring from BIP39 compatible seed mnemonics, you won't be able to get your MultiBit funds directly into CoinCorner (You'd have to use something like
Ian Coleman's Mnemonic Code Converter to convert the seed mnemonic to individual private keys then import/sweep those into CoinCorner)
The other alternative is to temporarily switch to Electrum (as per the videos from MultiBit devs for "
Classic->Electrum" and "
HD->Electrum", and then simply send your funds to your CoinCorner wallet (or whatever service/wallet you decide to use) from the Electrum wallet.
Personally, I'd recommend Electrum. Just make sure to get the official download from the official site:
https://electrum.org/#download