Is there any non-PC stand-alone alternative for translating vinyl to MP3?

Yes, but a properly configured PC will give you the most control over the process and give you the best results. For example, a PC can give you fine control over the recording levels using on-screen indicators, and allow you to perform normalization on the files so that the digital versions sound more similar in volume to the rest of your CD collection. A PC can also give you the ability to edit dust pops out of the recording and equalize the frequency response. It will also let you edit the starting and ending points of each track with great accuracy.

But if you're dead-set on not using a PC, then I would recommend buying a standalone CD recorder such as one of these. Once you have converted the LP into audio-CD format, you can do whatever you want with the Audio CD, such as play it on a CD player, rip it to MP3 on the PC, or put it into a Rio Central to convert it to MP3.

Of course, after having said all that...

If it were me...

I wouldn't even bother trying to convert any of the LPs to digital form at all. I would rather buy the LPs over again in CD format. Yes, it's expensive. Yes, it's irritating to think that I'm paying a record company twice to own a second copy of the same recording. But quality is of utmost importance to me, and when the record company masters the CDs, they are making them from the source master tapes, so the quality is much higher. And they will be mastered with an EQ and dynamic range that is appropriate for digital playback, rather than compressed and EQ'd to fit within the more limited dynamic range of a record player.

Of course, that doesn't count for the records in your collection that don't exist on CD (if any). But that's a whole different can of worms...