Yes. If you are using your mac (laptop), you can copy tracks from iTunes on Jooki. Open your Jooki dashboard website https://my.jooki.rocks, press the green "+" to create a new playlist, and use the drag and drop method as shown in the playlist creation tutorial here.
If file copying from iTunes on Jooki has no effect, it is possible that you are trying to copy files that have been "downloaded" under an Apple Music subscription. You can only copy tracks you have actually bought.
Please take your laptop, and check your files and filetypes in Finder/username/Itunes/ItunesMedia/...
Only m4a files can be dragged and dropped into jooki
If the files are audiobooks (.m4b), you might have seen that Jooki tries to reconnect after you drag the song into the browser. These type of files are encrypted with digital rights.
Why is that? In iTunes if you purchase an album, you own it and Apple gives (sells), you the rights to copy the files to your devices including Jooki (this right is limited, see iTunes' terms & conditions). With Apple Music they 'rent' you the files for as long as you keep paying for your Apple Music subscription. These files are protected against copy with Digital Rights Management (DRM).
iTunes not the only source for (children's) music. Click here to find out about "Where can I find music for Jooki?"