Is the fear of opening a website or an online company that might publish obscene pictures or clips, resulting in sin, considered a delusion (waswas) or an act of faith and caution to avoid falling into the forbidden?

Exaggeration (Ghulu) is reprehensible. And if piety (wara’) exceeds its bounds and leads to abandoning permissible things without a clear justification, then it becomes an excessive rigidity (tanattu’) and a blameworthy form of whispering (waswasa). The default rule for things is permissibility until a reason for prohibition becomes clear. Whoever understands this principle will ward off whispers from themselves. It is true that thehi who propagates what is forbidden bears the sin of those who disobey Allah through it. However, the scenarios you assumed are not inherent to websites; a person controls their website. Abstaining from opening websites for pages due to what you mentioned is not from praiseworthy piety.