We do not say 'to' after 'let'. However, your sentence is still incorrect because it is not possible to use 'let' in passive voice. Someone must let someone do something or let something happen. A correct option would be 'Cars are not allowed/permitted to ...'

Vehicles are not ALLOWED to pass through that area on Saturday night. We do not use "let" here.

'To' is very necessary. Actually, the word 'let' should be removed. The best ways to write that sentence would be: "Vehicles are not to pass through that area on Saturday night." or "Vehicles may not pass through that area on Saturday night."