Elliott Turnbull, chief executive of The Works on Kitson Road, Hunslet, said the raid could cost the charity up to £10,000.
Police said four young men who were wearing hoodies are being sought in connection with the raid, which happened overnight on Friday into Saturday morning.
The Works, which was first opened in 2003, was closed to the public at the weekend as the police investigated the incident
Mr Turnbull said burglars broke in through an external fire door and ransacked the premises.
He said they wrecked internal doors, smashed two tills and stole around £150 cash.
Also stolen were two laptop computers and two large monitor screens. Raiders also left a bank of 90 automated lockers unusable after destroying the computer which controls them.
Mr Turnbull said: “We had to turn away more than 350 people at the weekend, which is disappointing. We aim to make sure kids feel valued and supported so it’s disappointing that their peers have taken it away from them.”
Mr Turnbull said the incident will have caused a total loss to the charity of up to £10,000, taking into account items stolen, damage caused and loss of earnings.
The Works houses an indoor skate park, a gym, dance and fitness studio, classrooms, a climbing wall and large cafe area. The education centre aims to support young people who have not achieved their potential in mainstream education.
A West Yorkshire Police spokesman said: “Any witnesses or anyone who has any information is asked to contact officers at Elland Road via 101, quoting crime reference number 13160108695.”