Reconnnect fees may include physically re-enabling the utiliity as well as the cost of changing paperwork and electronic files related to your account.
The shutoff of utilities is often done on equipment owned by the utility rather than owned by the homeowner. Therefore you would need to get their permission to turn the utility back on, and they would be accepting liability in case you got hurt using their equipment or in case you damaged their equipment causing a leak or other problem. It is unlikely that they would allow this.