Make sure that the application being used supports the touchscreen, as some applications & menus do not use it.
If the touchscreen is supported by the application, please check the following:
Due to how PlayStation Vita's touchscreen works, you'll be unable to use the console with a stylus or whilst wearing gloves. If there is anything resting on the touchscreen please remove it and try again.
If a "screen protector" is being used, we recommended that you use an officially licensed product. Please visit the PlayStation Website for information on PlayStation Vita peripherals.
Once you've checked the above, please start the "Welcome Park" application to see if you can use the touchscreen normally.