Hello,
I run experiments written in Matlab and the Psychophysics Toolbox. I would like to run that software on the web via Inquisite 5. Is that possible? Thank you for any information! 

Inquisit cannot run experiments written in a different language, e.g Matlab's, just like Matlab cannot run experiments written in Inquisit.

What you would have to do is re-implement those experiment's in Inquisit's own language (whether that's possible depends on the specifics of those experiments, i.e. the type of stimulus manipulations etc. they perform).