You are facing this issue due to you have not assigned any Time off Policy to the Employee so that it shows empty in the Time off Type Dropdown in Add Time off Request Page.
Create a Time off Policy and assign the Policy type to the Employee. Please refer the below slideshare link which shows how to Create Time off Policy
Create A Time off Policy
Assign the Time off Policy type to the Employee in the Employee Detail Page (Employee–>Edit) Refer below Screen
Click on Edit as highlighted and you will be displayed with Employee Detail Page, select the Official Info Module & select the created Time off Policy from the Dropdown and click on Update.
By doing this process policy will be assigned to the employee & also you can see the Time off Type in the Dropdown in Add Time off Request Page.