Man, that's a challenge!!!
I think the hardest task at hand is to use your Apple IIc to copy the Apple II data files off your 5.25" disks, and somehow transmit the files out to something else via the Apple II's serial ports!
You have basically two (longshot) options; contact some dial-up service using a modem to make them available to an FTP server on the Internet—from there, you can use your Mac to fetch the files again and import them into AppleWorks 6 on the Mac; or, connect to another Mac that has serial ports using a cross-over serial cable.
For the first idea (via a dial-up service), you'll need a hosting or bulletin-board service that you can dial out to with a modem (because Apple IIc computers cannot use Ethernet networks!) so that you can send data to a terminal server using Apple II software such as ProTerm. That same provider will also need to make those files available via FTP so that you can fetch them with an application such as Interarchy or Fetch on the Macintosh.
I have not seen anyone offering such a service on the Internet (and on dial-up) anywhere in the world, but I haven't looked either—I suggest getting on board the A2Central Web site (
http://www.a2central.com/) or the comp.sys.apple2 newsgroup and start making enquiries there, to see if there is such a hosting service available to Apple II users.
If you don't have a modem for your Apple II, then all you have is the second idea (via serial port to a Mac), since there is no other way you can connect an Apple IIc to a Macintosh. Your Macintosh must have a serial port which can connect to the printer or modem port of the Apple IIc using a special cross-over cable designed for the purpose, and again, you'll need ProTerm in the Apple IIc to communicate to the Mac that is running something like Clarisworks Office 5 (the last version of AppleWorks for the Mac that supported communications terminals) or MacTerminal 3. The last Macintosh that was ever produced which had serial ports was the Power Macintosh G3 (beige) and that ran under Mac OS 9.2.1 as its latest (and last) system software release. Later model Macs must have a USB-to-serial port converter, and there are devices available from Keyspan (
http://www.keyspan.com/) and others which allow you to connect RS-232/422 serial devices to your USB-equipped Mac.
To add more complications, terminal software for the Mac only appeared for Mac OS 9... so you'll need to run Classic on a PowerPC-based Mac running up to Mac OS X 10.4 Tiger in order to run Macintosh-based terminal software. For an Intel-based Mac, you'll have to resort to using UNIX-based software to allow the Mac to communicate via any serial ports. I know of no software that can do this, and converter boxes may not have drivers that work on Intel Macs, so the chances of this idea actually becoming useful is rather slim!
As you can see, this is a pretty big project, either way you go! In a nutshell, to transfer Apple II data to a Mac (and vice versa) you need:
--an Apple II with a modem, and a dial-in service provider that can accept dial-up connections from Apple II communications software, and publish files that you upload to an FTP, HTTP or WebDAV server on the Internet (best solution for Mac OS X users), or
--an Apple II, a Power Macintosh G3 or earlier running Mac OS 9*, a cross-over serial cable between the Apple IIc and the Macintosh serial ports, and communications software running on both machines for transferring files from one machine to the other.
[* Or a later-model Mac with a USB-to-serial port converter device on Mac OS X 10.4 Tiger.]
I'd speak to some people over at A2Central or others at comp.sys.apple2 (on Usenet—use the Macintosh application Unison or similar, or a Web service that allows you to correspond via NNTP news) in order to find out if such services exist. If you ask nicely, you might also have someone set up such a service for you, because frankly, I think the Apple II community would find such a service indispensable.
I know this is not a solution, but this is the best I can describe as options for transferring Apple II files to a Mac, and I have the feeling you need more hardware or services than you have.
A couple more comments... and I am afraid I am clutching at straws again... you could see if anyone has or is selling a Macintosh LC with an Apple IIe Card and an Apple 5.25 drive, a monitor and keyboard & mouse. That machine would be ideal for transferring the data you want over to Macintosh disks or to a Macintosh network. Another idea could be to get a UniDisk 3.5 drive for the Apple IIc, and a 3.5" drive for a USB-based Macintosh that can run Classic, and transfer the data from 5.25" disks onto 3.5" disks on the IIc which you can then transport to the Mac—the problem here is that Mac OS X cannot read ProDOS disks, since there are no file system translators for that filesystem on any version of Mac OS X. Again, you'll also need an older Mac running Mac OS 9 to do this.