Can Teensy 3.2 \ 3.1 be used as USB host device without additional shields in order to control some other USB device?
Check out which IC your board is featuring, then refer to this page and download the corresponding datasheet. If it mentions USB OTG or USB host, then the answer is yes, otherwise no.
It seems that Teensy 3.1 & 3.2 are based on MK20DX256, which does support OTG. Note that it is "full speed", meaning you're limited to 12Mbps and devices which require higher transfer rates may not work.