Yes it does. In fact, lack of sleep can even cause anxiety in people who don't normally experience such condition. It makes things worse for people already living with anxiety.
Sleep debt doesn't just directly cause anxiety. It can lead to other problems that may worsen anxiety. For example, sleep deprivation causes brain stress that can lead to further anxiety. Lack of sleep may also cause body stress that could have been relieved during sleep.
So have plenty of sleep if you start to experience anxiety that don't usually occur. But remember that only one night of good sleep is not going to solve the problem. You may have to get enough sleep for several weeks in a row before you recover fully.
Keyword: lack sleep anxiety