My Battlefield 5 weapon is level 10 but no mission was assigned.

Hi yull20070812​,

If the assignment is not in the Armory waiting to be claimed, you can try the suggestions in the "Item Acquisition" section of the BFV FAQ:

Battlefield V FAQ | EA Forums - 6825895 

However, if those do not work either, you will need to visit help.ea.com and open a support case.  

Just to let you know, missing items cannot be granted through these forums, as nobody has access to game accounts for security reasons.