Can anxiety cause fasiculations?

Yes. So can caffeine. Typically the small muscles around the eye are affected most. If fasciculations persist after reducing stress and caffeine- see a neurologist. (hemifacial spasm can also cause involuntary muscle movement in the face.).
Fasciculations. Small muscle twitching is mostly benign, but some causes can include magnesium deficiency, benzodiazepines withdrawal, anticholinergic drugs such as Benadryl (diphenhydramine) over long period of time, caffeine, stimulants used to treat adhd, & asthma medications such as proventil.