They are called お絞り, typically written in Kana as おしぼり.
The word お絞り comes from the verb 絞る meaning "to wring", with the honorific prefix お.
If you want to be more specific:
布製おしぼり are made from cloth.
紙製おしぼり are made from paper (these are the disposable, plastic packaged ones)
'To vape' is 電子タバコを吸う
The noun 電子タバコ literally means electronic cigarette and the verb 吸う has many meanings aside (not just smoking), it can also mean to inhale, breathe, suck etc.
This Japanese game (はって言うゲーム) has a card that shows how versatile the 吸う verb is (funnily enough, none of which involve smoking).