@a8ree1 - Here is a ready solution for this:
1.Your client will copy the CSV files from local machine to Azure Storage using AzCopy.
2.At your end, you can use Azure Data Factory to create a pipeline in order to copy that data from storage to MySQL database once or repeatedly on schedule.
The second link above have all instructions and a button to directly deploy a pipeline to accomplish this task. Please let us know if you have any questions.
Regards
Navtej S