Have been noticing this within last month or so. Has at least been happening after match completion, usually back at home page of client. Only solution has been to force close in the task manager. Was able to go through the store, my profile, even forge a key to open a new hextech chest. Only issue is the close X will not respond. It does highlight and show the tag "click to close" and the Minimize button works. Only close X has been having the issue. To the best of my knowledge all my drivers are up to date.
Windows 10, 64 bit Home
32 Gig Ram
AMD FX 8350 - 4 Ghtz - 8 Core
Nvidia GeForce GTX 760 by Asus
Samsung SSD 850 Pro 256 Gig