Making mp3 with meta data embedded
The quickest way to do this for just one file, or a short album of files is to use iTunes, it’s cross-platform and behaves fairly consistently.
First, create your .wav file, it should be no higher in SR than 48k and 24bit for iTunes to like it.
Open iTunes and setup your exporting/converting settings.

Setup Import Settings

Choose the mp3 encoder and setup settings like this

When your encoding is setup, drop your .wav file into iTunes and select the file, then in the file menu, select convert, “Create mp3 version” should show up:

When it’s converted, select the newly minted mp3 file and right click on the file and select album info, there, you can type in all of your meta data and drop in your artwork.
When that’s done, select the file again and go to the file menut and choose “Show in Finder”, this takes you to the location of the mp3 file. Select it and copy it into your submission directory, all of the art work and meta data will have been embedded into the file and will travel with it.

If you don’t want to use iTunes, there are plenty of other ways to embed metadata into mp3s, when you export it from Logic as .mp3, you can add iD3 tags, similarly with Adobe Audition, possibly with pro-Tools and potentially with Reaper. Also there are commandline tools galore and dedicated apps such as these for windows and OSX respectively.