Applications are installed into /var/lib/install to not mess up with the rest of the system. It should be possible to make this a symlink pointing to somewhere on the MMC. The filesystem on the MMC, however, is a dumb FAT and the card is mounted with noexec flag.

I did not try it, but I'm sure that by reformatting the MMC and modifying /etc/fstab, you could install applications on the MMC.