Having grafted a few mulberries, yes, it is possible. However it is a bit tricky, as mulberry produces a lot of sap. Some grafting techniques and times might drown the scion. I have grafted successfully in pots, before they go into vegetation.
There is also a technique I haven't tried, which is said to be specifically for grafting mulberries without drowning them.
Here one of the pictures:
Here the complete instructions with more pictures, use anything to translate:
https://wiki.fruitiers.org/index.php/Greffe_en_gueule_de_brochet